Diploma thesis

Towards a pure heralded
single photon source
Diploma thesis
Andreas Christ
Integrated Quantum Optics Group
Max Planck Junior Research Group at the Max Planck Research Group, Institute of
Optics, Information and Photonics
University Erlangen-Nuremberg
August 2008

Contents
1 Overview 1
2 Introduction 3
3 Theory 5
3.1 Three-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.1 Sum frequency generation . . . . . . . . . . . . . . . . . . . . 5
3.1.2 Parametric down conversion . . . . . . . . . . . . . . . . . . . 7
3.1.3 Engineering the PDC process . . . . . . . . . . . . . . . . . . 11
3.2 Pure heralded single photons . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Quantum networks and heralding . . . . . . . . . . . . . . . 21
3.2.2 Frequency entanglement of two-photon-states . . . . . . . . . 24
3.3 Generating pure heralded single photons . . . . . . . . . . . . . . . . 26
3.3.1 Spectral filtering . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Group velocity matching . . . . . . . . . . . . . . . . . . . . . 29
3.3.3 Counterpropagating signal and idler fields . . . . . . . . . . . 33
3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Experiment 41
4.1 Avalanche photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 id Quantique id201 APD . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Dark counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Afterpulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.3 Remaining pump light in PDC detection . . . . . . . . . . . . 44
4.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Characterization of the waveguide chips . . . . . . . . . . . . . . . . 46
4.3.1 Chip BCT0703-B12 . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Chip ITI0706-B12 . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 PDC experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Conclusion and Outlook 63
A Appendix 65
A.1 Mathematica packages . . . . . . . . . . . . . . . . . . . . . . . . . . 65
i

ii
A.2 id201 programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.3 APD characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3.1 Dark counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3.2 1310 nm cw laser . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.3.3 1555 nm cw laser . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.3.4 808 nm pulsed laser . . . . . . . . . . . . . . . . . . . . . . . 113
A.4 SHG data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.4.1 Different SHG processes in KTP . . . . . . . . . . . . . . . . 115
A.4.2 Chip BCT0703-B12 . . . . . . . . . . . . . . . . . . . . . . . 116
A.4.3 Chip ITI0706-B12 . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography 139

1 Overview
In the scope of this thesis, we theoretically and experimentally investigate the spatial
and spectral properties of three-wave mixing processes. We extend the theoretical
framework of waveguided parametric downconversion (PDC) to included different
spatial modes and evanescent waves in the surrounding material. We compare the
fidelity of different approaches to generate pure heralded single photon states. This
is an important prerequisite for further PDC experiments. To perform all these
calculations we developed a simulation which can deal with all needed formulas
to describe the PDC process and analyse the spectral correlations by a Schmidt
decomposition.
Different waveguide samples have been examined in the laboratory with second
harmonic generation (SHG). The gained information proved useful to construct further
experiments with these crystals. Additionally, this data yielded the possibility
to test the performance of the theoretical framework in comparison to experimental
data. Finally, we observed PDC photon pairs taking into account the measured
crystal properties.
1

Chapter 1. Overview
2

2 Introduction
The reason why truth is so much
stranger than fiction, is that there is no
need for it to be consistent.
Mark Twain
Quantum mechanics has been one of the most important paradigm change in physics
to this day. Initiated by Max Planck with his hypothesis of quantised black body
radiation in the year 1900 and Einsteins explanation of the photoelectric effect 1905,
the photon theory was one of the first achievements of quantum theory. Yet it took
over 50 years for the first undeniable proof of its existence by Kimble et. al. 1977
[1].
It is safe to assume that neither Einstein nor Planck, at the beginning of the
20th century, ever considered the generation of single photons. However, the advent
of quantum information and the rapid progress in quantum information processing
with linear optical gates [2] and cluster states [3] has placed high demands on
the generation of single photons. Required are pure deterministic or at least pure
heralded single photon states.
At the dawn of the 21st century, single photon sources of all kind have emerged.
They came a long way from the first experiments with strongly attenuated lasers
beams some decades ago. They range from quantum dots in pillar microcavities [4],
falling neutral atoms [5], trapped ions in cavities [6], defects in diamond nanocrystals
[7], a single molecule in solid [8], four-wave mixing [9], to parametric downconversion
[10, 11, 12]. They all display completely different approaches to the same goal,
explored by scientists around the world.
Until recently, it has been very difficult to generate the desired pure single photon
states. In October 2007, a new type of pure heralded single photon sources has
been presented [10]. The authors rely on parametric downconversion in bulk KDP,
pumped by a ultrashort laser source.
In this thesis, we take a deep dive into this thrilling field of nonlinear optics
and theoretically investigate different methods to implement similar single photon
sources. We push this approach forward, with ideas for a higher brightness and
a more straightforward implementation in KTP. Waveguiding structures embedded
in nonlinear crystals are a perfect candidate. They exhibit a much higher
brightness than bulk crystals. We explore the possibility to shift the wavelength of
the generated photons up to the telecommunication wavelength of 1550 nm, where
propagation loss through optical fibers is minimal. On the experimental side, we
characterised different sample waveguides in the laboratory with second harmonic
generation. The results enable us to compare the developed theory with obtained
3

experimental data.
This thesis is divided into three parts. In the first part, we give an introduction to
three-wave-mixing processes, especially second harmonic generation and parametric
downconversion. We proceed with a discussion of the encountered challenges
and different approaches to create pure heralded single photon states. They range
from common methods used in the lab to techniques that still require technological
progress. In the experimental Section, we turn to the problem of detecting single
photons. This is followed by a characterisation of our waveguides with the observation
of second harmonic generation. We gain deep insight into the properties,
of the waveguides and compare the theoretical predictions with experimental data.
Finally, we present the first observation of parametric downconversion in our crystal
with promising results for the future.
4

3 Theory
3.1 Three-wave mixing
Nonlinear optics is the science of light-matter and light-light interaction in nonlinear
materials. Effects are ”nonlinear” in the sense that the response of the material
depends nonlinearly upon the applied electric field. Interactions of this kind are
embedded in a wide range of phenomena, from the Kerr-effect to frequency mixing
processes. In this work, we focus solely on three-wave mixing processes.
3.1.1 Sum frequency generation
Since its discovery in 1961 by Franken et al. [13], second harmonic generation
(SHG) or sum frequency generation (SFG) became one of the most commonly used
nonlinear optical effects. Its use ranges from tiny green laser pointers to expensive
mode locked pulsed Nd:YAG laser systems. SFG describes the combination of two
photons to a third photon with higher energy, inside a nonlinear material, under
the restrictions of energy and momentum conservation. In the special case of two
identical photons merging, this process is called SHG (see Figure 3.1). The classical
Figure 3.1: Sum frequency generation, energy conservation and momentum
conservation
differential equation describing SFG is derived from the Maxwell equations assuming
5

a nonlinear, nonmagnetic material, that is free of charges and currents:
−∇ 2 � E + 1
c 2
∂2 ∂t2 � E = − 4π
c2 ∂2P� . (3.1)
∂t2 For dispersive nonlinear materials and three interacting waves, this equation can be
solved with a quasi-monochromatic Ansatz, and yields the intensity of the upconverted
field [14]:
I3(ω3) = (2π)524χ (2) 2
123 I1(ω1)I2(ω2)
n1n2n3λ2 3c L 2 sinc 2
� �
∆kL
. (3.2)
2
Equation 3.2 describes the coupling between the three involved fields. The efficiency
of SFG depends on the momentum mismatch ∆k, between the interacting waves,
where we assume strictly collinear waves propagation.
∆k = k3(ω3) − k2(ω2) − k1(ω1). (3.3)
Only if the combined momentum of all involved photons equals zero an efficient SFG
is possible. The Sellmeier equations or refractive indices for the crystal therefore have
to fulfil Eq. 3.4 under the restrictions of energy conservation (ω3 = ω2 + ω1).
n1ω1 + n2ω2 = n3ω3
(3.4)
The coupling constant of the three interacting waves with different polarizations,
in SFG, is given through the χ (2)
ijk-tensor. In lossless crystals, this tensor can be
reduced to a two dimensional matrix dij because of crystal symmetries [14].
⎛ ⎞
⎛
⎝
Px
Py
Pz
⎞
⎛
⎠ = ɛ0 ⎝
d11 d12 d13 d14 d15 d16
d21 d22 d23 d24 d25 d26
d31 d32 d33 d34 d35 d33
(Ex) 2
(Ey) 2
(Ez) 2
⎞ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎠ · ⎜ ⎟
⎜
⎜2EyEz
⎟
⎝2EzEx
⎠
2ExEy
(3.5)
The fields EiEj on the right hand side of equation 3.5 are the incoming pump fields.
They introduce a nonlinear polarization Pi, that emits a SFG field with the same
polarization, but a different frequency. In common nonlinear materials numerous
coupling constants dij are close to zero, and only a small number of possible threewave-interactions
remains. In the case of orthogonally polarized incoming pump
fields, this is called type-II SFG. Identically polarized pump waves are denoted by
type-I SFG.
Equation 3.2 still lacks the consideration of polychromatic waves, common for
pulsed laser sources. Hence, we generalize equation 3.2 to describe one broadband
laser pulses propagating through the crystal. The pulse exhibits a Gaussian shape
in the frequency domain, as follows:
6
I ′ (ω−ωp)2
−
2σ
p(ω) = Ipe 2 p (3.6)

Applied to Eq. 3.2 (I1 = I2 = I ′ p(ω)), this requires an integration over the pump
spectrum, to consider all different mixing possibilities for one particular output
SFG frequency. For the sake of a clear arrangement, we merged all slowly varying
variables into the factors K and K’:
� ∞ � ∞
I3(ω3) = dω1
0
� ∞
=
0
0
dω2K ′ I 2 pe − (ω 1 −ωp)2
2σ 2 p e − (ω 2 −ωp)2
2σ 2 p sinc 2
dω1KI 2 pe − (ω 1 −ωp)2
2σ 2 p e − (ω 1 −ω 3 −ωp)2
2σ 2 p sinc 2
� �
∆kL
2
� �
∆kL
2
(3.7)
The generated SHG pulse exhibits a Gaussian shape similar to the input wave as
shown in Figure 3.2. Equation 3.7 will prove useful to predict our SHG results.
Figure 3.2: Pump spectrum and corresponding upconverted SHG spectrum
3.1.2 Parametric down conversion
Parametric down conversion (PDC) can be regarded as the reversal of SHG. In this
process, an incoming pump photon decays inside a crystal with a χ (2) -nonlinearity
into two photons that are called signal and idler, for historical reasons (Figure 3.3).
PDC was first predicted theoretically by Klyshko in 1968, and experimentally observed
in 1970 by Burnham and Weinberg [15]. In recent years, it became a common
source for the generation of entangled photon pairs.
The Hamiltionian of the PDC process can be derived from the energy density in
a nonlinear medium:
ˆH = χ (2)
�
V
dV Ê(+)
p
With electric field operators defined as [16]:
Ê (−)
s Ê (−)
i + h.c.. (3.8)
Ê (−)
� �
∞ �ωµ
µ (�r, t) = dωµ
0 2ɛ0V e−i(� kµ(ωµ)�r−ωµt) †
â µ (ωµ) . (3.9)
7

Figure 3.3: Parametric down conversion, energy conservation and momentum
conservation
Ê (−)
� � †
(+)
µ (�r, t) = Ê µ (�r, t)
(3.10)
Analogous to SFG, the χ (2) -tensor element describes the coupling between different
polarizations involved. Yet, the process is reversed. The polarizations on the
left hand side of equation 3.5 are created by the incoming pump field carrying the
identical polarization. The fields on the right hand side describe the polarizations
of the emitted signal and idler fields [17]. Orthogonally polarized signal and idler
fields are called type-II PDC, identically polarized downconverted fields are named
type-I PDC.
To describe the process of PDC in detail, we have to compute the time evolution of
an input state under the exposure of the Hamiltonian in Equation 3.8. The evolution
of a quantum state can be calculated with the Schrödinger equation.
� � t 1
|ψ (�r, t)〉 = exp dt
i�
′ �
H(�r, ˆ ′
t ) |ψ (�r, t0)〉 (3.11)
0
In order to solve this equation, we expand the exponential function into a Taylor
series:
|ψ (�r, t)〉 =
�
1 + 1
� t
dt
i� 0
′ �
H(�r, ˆ ′
t ) + · · · |ψ (�r, t0)〉 . (3.12)
Terms of order two and higher, are responsible for the creation of multiple photon
pairs, procreated by the annihilation of multiple pump photons (âpâpâ † sâ †
i ↠s⠆
i ). The
PDC interaction is weak, therefore the creation of multiple photon pairs is negligible,
in the scope of this thesis. Consequently, we are only focusing on the creation of
one photon pair by the destruction of one incoming pump photon. We assume no
pump depletion and treat the strong pump field as a coherent wave with a Gaussian
frequency distribution αp(ω):
� ∞
Ep (�r, t) =
0
dωαp(ω)e −i(� kp(ω)�r−ωt) + c.c. (3.13)
.We assume a pulsed laser system that creates Gaussian pulses in the frequency
domain, with a given pump central frequency ωp and a pump width σ, according to
8

the rotating wave approximation [18]:
(ω−ωp)
−
αp(ω) = Ape 2σ (3.14)
By substituting Equations 3.9, 3.10, 3.13 and 3.14 into 3.8, we obtain for the Hamiltonian:
ˆH (�r, t) = χ (2)
�
dV
� ∞ � ∞ � ∞
(ω−ωp)
−
dω dωs dωiApe 2σ As(ωs)Ai(ωi)
V
0
0
0
e i(� kp(ωp)− � ks(ωs)− � ki(ωi))�r e −i(ωp−ωs−ωi)t â † s (ωs) â †
i (ωi) + h.c. (3.15)
.The parameters for the envelope of signal and idler frequencies have been merged
into As,i.
We now assume that our initial state |ψ(t = 0)〉 is the vacuum state |0〉. The
corresponding two-photon-state |ψs,i〉 evaluates to:
|ψs,i〉 = |0〉 +
� 1
i� χ(2)
psi
� ∞ � ∞ � ∞
0
0
0
�� t
(ω−ωp)
− 2σp dω dωs dωiApe As(ωs)Ai(ωi) dt
0
′ e −i∆ωt′
��
·
V
dV e i∆� �
k(ωp,ωs,ωi)�r
â † s (ωs) â †
i (ωi)
�
+ h.c.
(3.16)
The functions As,i and Ap are slowly varying with frequency, we hence treat them
as constant and combine them with the susceptibility into the constant A ′ . Furthermore,
the leading vacuum term is of no particular interest to us and will therefore
be omitted in future calculations. The hermitian conjugate part covers the reverse
process. Since only few photons are downconverted the reversal process is neglected.
This leads to the following state:
|ψs,i〉 = A ′ V
� ∞ � ∞ � ∞
0 0
� �
1
·
V
0
�� t
dω dωs dωiα(ω)
dt
0
′ e −i∆ωt′
V
dV e i∆� �
k(ωp,ωs,ωi)�r
â † s (ωs) â †
i (ωi) |0〉 . (3.17)
In this formula ∆k represents the momentum mismatch already introduced in the
SHG treatment (Equation 3.3).
∆k = kp(ωp) − ks(ωs) − ki(ωi) (3.18)
We extent the integration time to ±∞, since we are interested in the steady state.
With this simplification, the time integration is easily solveable and yields:
� ∞
dt ′ e −i∆ωt′
= 2πδ(∆ω) (3.19)
−∞
The result is the well known energy conservation condition, with ∆ω = ωp − ωs − ωi,
that replaces the frequency mismatch. We employ the energy conservation condition
to get rid of the ω integration.
�
9
�
|0〉

For the volume integration, we restrict ourselves to a strictly collinear propagation
of the pump, signal and idler beams through the crystal (see Chapter 3.1.3 for
details). The integration over the crystal length yields the formula:
1
V
�
V
dV e i∆� k(ωp,ωs,ωi)�r → 1
L
� L
dze i∆k(ωp,ωs,ωi)z
�
∆kL
= sinc
2
0
Finally, we obtain the following description of the two-photon-state:
|Ψs,i〉 = 2πA ′ � ∞ � ∞
L dωs dωie − (ωs+ωi−ωp)2 2σ2 �
∆kL
sinc
2
0
0
�
∆kL
i
e 2 (3.20)
�
∆kz
i
e 2 â † s (ωs) â †
i (ωi) |0〉
(3.21)
For further discussion, we focus on the spectral properties of PDC. The phase contributions
are not considered in this thesis:
� ∞ � ∞
|ψs,i〉 = A dωs dωie − (ωs+ωi−ωp)2 2σ2 � �
∆kL
sinc â
2
† s (ωs) â †
i (ωi) |0〉 . (3.22)
0
0
Furthermore, we approximate the sinc contribution with a Gaussian distribution
(sinc(x) ≈ exp � −γx2� with γ = 0.193 . . .).
� ∞ � ∞
|ψs,i〉
y,z = A
0
0
dωs dωie − (ωs+ωi −ωp)2
2σ2 ∆kL
−γ(
e 2 )2
â † s (ωs) â †
i (ωi) |0〉 (3.23)
As one can readily verify the spectral contributions of the two-photon state are given
by two Gaussian functions. The first one,
α(ωs + ωi) = e − (ωs+ω i −ωp)2
2σ (3.24)
is named the pump envelope. It contains the pump parameters, pump central frequency
ωp and pump width σ. It implies the energy conservation condition. The
second Gaussian distribution,
� �
∆kL
φ(ωs, ωi) = sinc
2
∆kL
−γ(
≈ e 2 )2
(3.25)
is referred to as the phasematching function and ensures momentum conservation.
In SFG and PDC the same formula occurs (3.3 and 3.18) i.e., both processes share
the same phasematching.
Both pump envelope and phasematching function constitute the joint spectral
amplitude (JSA) of the PDC-created two-photon state:
f(ωs, ωi) = e − (ωs+ωi−ωp)2 ∆kL
−γ( 2σ e 2 )2
Finally, we write the two-photon state in a compact notation:
� ∞ � ∞
|ψs,i〉 = A
10
0
0
(3.26)
dωs dωif(ωs, ωi)â † s(ωs)â †
i (ωi) |0〉 . (3.27)

Figure 3.4: Pump envelope x phasematching function = JSA
To understand the physical meaning of α(ωs + ωi) · φ(ωs, ωi) = f(ωs, ωi) it is very
helpful to visualise these functions in the {ωs, ωi}-plane as plotted in Figure 3.4.
The pump distribution always exhibits a negative slope of -45 ◦ . Its position is
given by the pump frequency ωp and its width by the pump width σ. The form
of the phasematching function depends on the type of downconversion. In a type-I
downconversion, the phasematching function is axially symmetric to the +45 ◦ slope,
in contrast to a type-II downconversion process. In this scenario, the phasematching
contour exhibits no symmetries.
Together they always form a two-dimensional Gaussian distribution in the frequency
domain.
3.1.3 Engineering the PDC process
PDC as a source of quantum states suffers from two main drawbacks. Firstly, the
phasematching function φ(ωs, ωi) depends only on the Sellmeier equations of the
crystal, so it is not possible to generate photon pairs at arbitrary wavelengths.
The contour of the phasematching slope is given solely by the crystal properties.
Secondly, the conversion efficiency is in the range of 10 −10 , signal and idler intensities
are located at the single photon level. In the past, different proposals have been made
to overcome these restrictions. In this Section, we present two of them: Waveguiding
structures and periodical poling.
Periodical poling
Up to now, the nonlinearity of the material has been assumed to be independent
of spatial variation. Let us consider a crystal with a periodically varying nonlinear
index along the propagation direction.
With current technology it is possible to create a step index change of the nonlinearity
by applying strong electric fields during the production process. The result is
sketched in Figure 3.5. The period of this index change is called grating period and
denoted Λ. Established grating periods are in the range of micro meters.
11

Figure 3.5: 1D square-wave periodical poling period
We have to treat this varying nonlinear index by introducing spatial variations in
the previously constant χ (2) . The new nonlinear index is modelled as a square wave
function:
χ (2) (z) = χ (2) � � ��
2πz
sgn sin . (3.28)
Λ
The physical meaning of this function can be understood if we transform Formula
3.28 into a Fourier series:
χ (2) (z) = χ (2) �
∞�
�
1 2πm π
x+i
Re ei Λ 2 m = ±1, ±3, ±5 . . . (3.29)
m
m=−∞
.As in the previous discussion, we neglect the phase factor and the PDC Hamiltionian
is altered in respect to Eq. 3.23:
|ψs,i〉 = A
∞�
m=−∞
1
m
� ∞ � ∞
0
0
dωs dωie − (ωs+ωi−ωp)2 2σ2 e −γ
�
(kp−ks−ki − 2πm
Λ )L
2
� 2
â † s (ωs) â †
i (ωi) |0〉 .
(3.30)
Hence the grating period introduces an additional term in the momentum mismatch,
named quasiphasematching vector kQP M = 2πm
Λ . The momentum conservation
is modified:
∆k = kp − ks − ki − kQP M
(3.31)
The influence of a grating period is visualized in Figure 3.6. The phasematching
function gets shifted in the frequency plane. That way, periodic poling is a very
powerful tool to generate photon pairs at, in principle, arbitrary frequencies. This
has been applied, in the laboratory, to produce degenerate downconverted photon
pairs around 800 nm, where efficient detectors are available, or at the telecommunication
wavelength of 1550 nm. At this frequency, the transmission loss through
optical fibers is minimal.
12

Figure 3.6: Different Λ Figure 3.7: Higher order phasematching
Eq. 3.30 also reveals that the square wave grating orders provide several phasematching
curves, m = ±1 as the two main periods and several higher order grating
periods m = ±3, ±5, ±7 . . . some of which are visualized in figure 3.7. However the
higher order modes are suppressed by 1
m
in amplitude. Additionally, these higher
order phasematching modes soon drift into regions, where the produced photons
can’t be detected any more.
Waveguides
Waveguiding structures as depicted in Figure 3.8 exhibit two main advantages over
bulk crystal. Firstly, the propagation of the signal, idler and pump beams is now
Figure 3.8: Embedded 2D waveguiding structure
strictly collinear. This restricts the momentum mismatch to one dimension, as already
assumed in Equation 3.23. Because of the collinear propagation of all electric
fields, this two-photon source is now much more convenient to handle experimentally.
In contrast to bulk crystal, where angular spread, which is a result of angular
phasematching, has to be considered. Secondly, the pump beam undergoes a high
modal confinement in waveguides. These constrictions lead to a high photon flux
of signal and idler in a narrow solid angle. The conversion efficiency from pump
photons to downconverted photons is increased by several orders of magnitude. The
waveguides restrict the fields to certain well defined spatial modes (see Figures 3.9
13

and 3.10).
Figure 3.9: 0-0 waveguide mode Figure 3.10: 0-1 waveguide mode
In geometrical optics, the light rays are not travelling straight through the waveguide,
but are reflected up and down from left to right inside the guiding structure
(see Figure 3.11). Therefore, we cannot assume a linear propagation of the waves
through the crystal like in our derivation in Equation 3.23. In a 2-dimensional
waveguide, with perfect conducting edges and width and height b, it is possible to
consider these effects by introducing an effective refractive index [19].
neff ≈
�
nbulk − 2
� �2 λ
2b
(3.32)
However, this model is rather crude. It neither considers evanescent waves nor wave
propagation in the bulk crystal nor higher order spatial modes (Figure 3.12).
Figure 3.11: Light propagation Figure 3.12: Different spatial modes
For a consideration of these effects, we have a more advanced model [19] available,
that has already been implemented [20]. It accounts for evanescences in the
surrounding material and transversal modes. The differences, in refractive indices
between bulk crystal and slab waveguide and a waveguide with evanescences are not
negligible (Figure 3.13).
This model predicts different refractive indices for different spatial modes (Figure
3.14). That way multiple spatial modes in the waveguide introduce several slightly
14

1.90
1.85
1.80
1.84
1.82
1.80
1.78
1.76
neff
neff
Modelling
advanced
2.5 3.0 3.5 4.0 4.5 5.0 Ω�PHz�
Figure 3.13: Comparison between neff for bulk crystal, slab waveguide
and the model considering evanescences
3 4 5 6 Ω�PHz�
Figure 3.14: neff in waveguides
�0,0�
�0,1�
�1,1�
0.010
0.008
0.006
0.004
�neff
bulk
simple
�0,0���0,1�
�0,0���1,1�
3 4 5 6 Ω�PHz�
Figure 3.15: neff difference in waveguides
shifted phasematching contours (Figure 3.18). The differences in effective refraction
index increase significantly for lower frequencies (Figure 3.15). For PDC this has
higher influence on the signal and idler modes than on the pump modes (Figure
3.15).
With the expansion of our model to higher order spatial modes, modal overlap
between signal, idler and pump modes has to be taken into account. It will limit the
efficiency of this ”higher order” phasematching. The transverse spatial dimensions
of the electric fields have been neglected in the prior derivation of PDC (Eq. 3.23).
15

By inserting the corresponding terms into the two-photon-state we obtain:
� ∞ � ∞ � b � b
|Ψs,i〉 = A dωs dωi dx dyEp(x, y)Es(x, y)Ei(x, y)
0
0
0
0
e − (ωs+ωi−ωp)2 2σ2 ∆kL
−γ(
e 2 )2
â † s (ωs) â †
i (ωi) |0〉 . (3.33)
The waveguiding structures in our laboratory are tooth shaped as shown in Fig.
3.16. To calculate the modal overlap we approximate this waveguide as a twodimensional
rectangular structure of width and height b, with perfect conducting
edges, as depicted in Fig. 3.17.
Figure 3.16: Real waveguide (50x) Figure 3.17: Assumed 2D waveguide
The transversal modes for this scenario are:
�
nπ
Eµ(x, y) = Aµsin
b x
� �
mπ
sin
b y
�
n, m ∈ N\{0}. (3.34)
Equation 3.33 is separable, i.e. E(x, y) = E(x)E(y), and leads to two independent
integrals.
� b � b
dx dyEp(x, y)Es(x, y)Ei(x, y)
0
0
� L
� L
= dxEp(x)Es(x)Ei(x) dyEp(y)Es(y)Ei(y) (3.35)
0
We solve this one dimensional integration as follows:
� b �
nπ
dx sin
0 b x
� �
mπ
sin
b x
� �
oπ
sin
b x
�
= b
4π (
1
1
cos(π(n + m + o)) +
cos(π(−n + m − o))
n + m + o −n + m − o
1
1
+ cos(π(n − m − o)) +
cos(π(−n − m + o))) (3.36)
n − m − o −n − m + o
16
0

In this solution, there are four selection rules that restrict the mode coupling:
n + m + o �= 0
−n + m − o �= 0
n − m − o �= 0
−n − m + o �= 0
n, m, o ∈ N\{0}. (3.37)
We evaluated these up to n = m = o = 3 and obtained 18 different possibilities with
subsequently normalised three-wave coupling factors (see Table 3.1). In the table,
pump mode → signal mode + idler mode coupling constant
n-1 → m-1 + o-1
0 → 0 + 0 1.00
1 → 1 + 1 0.50
2 → 2 + 2 0.33
0 → 1 + 1 0.80
1 → 0 + 1 0.80
1 → 1 + 0 0.80
2 → 0 + 0 0.20
0 → 2 + 0 0.20
0 → 0 + 2 0.20
0 → 2 + 2 0.77
2 → 0 + 2 0.77
2 → 2 + 0 0.77
2 → 1 + 1 0.57
1 → 2 + 1 0.57
1 → 1 + 2 0.57
1 → 2 + 2 0.42
2 → 1 + 2 0.42
2 → 2 + 1 0.42
Table 3.1: Some mode coupling possibilities in 1D waveguides
we switched into the standard notation for fiber and waveguide modes starting with
(0,0). This is a more common notation in the field of optics and is usually used to
label guided modes in fibres.
We expand our findings to two dimensional waveguides and predict 324 different
mode couplings, some of which are depicted in Table 3.2. All of these possibilities
corresponds to a different set of Sellmeier equations that have to be applied. They
only differ slightly, but all produce distinguishable phasematching contours (see
Figure 3.18).
We already observed higher order spatial mode waveguided parametric downconversion.
We measured signal and idler spectral distributions from several simultane-
17

Λi
phasematching
Λs
Figure 3.18: Different predicted higher order spatial mode phasematching
contours
ously excited JSAs, as plotted in Figure 3.19 and 3.20. These measurements are in
good qualitative agreement with our modelling, yet it is difficult to find a quantitative
accordance. The waveguides are not rectangular and the pump pulses incident
into the crystal don not show the assumed properties. Hence further investigations
in theory and experiment have to be made.
18

signal power [a.u.]
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
Signal Spectrum
0.3
650 700 750 800 850 900 950
signal wavelength [nm]
Figure 3.19: Measured signal distribution
Signal intensity �a.u.�
5
4
3
2
1
750 800 850 900 Λ�nm�
Figure 3.21: Predicted signal distribution
idler power [a.u.]
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
Idler Spectrum
0.3
650 700 750 800 850 900 950
idler wavelength [nm]
Figure 3.20: Measured idler distribution
Idler intensity �a.u.�
3.5
3.0
2.5
2.0
1.5
1.0
0.5
750 800 850 900 Λ�nm�
Figure 3.22: Predicted idler distribution
19

pump mode → signal mode + idler mode coupling constant
(n1 − 1)(n2 − 1) → (m1 − 1)(m2 − 1) + (o1 − 1)(o2 − 1)
(0,0) → (0,0) + (0,0) 1.00
(0,1) → (0,1) + (0,1) 0.50
(0,0) → (0,1) + (0,1) 0.80
(0,1) → (0,0) + (0,1) 0.80
(0,1) → (0,1) + (0,0) 0.80
(0,2) → (0,0) + (0,0) 0.20
(0,2) → (0,1) + (0,1) 0.42
(1,0) → (1,0) + (1,0) 0.50
(1,1) → (1,1) + (1,1) 0.25
(1,0) → (1,1) + (1,1) 0.40
(1,1) → (1,0) + (1,1) 0.40
(1,1) → (1,1) + (1,0) 0.40
(1,2) → (1,0) + (1,0) 0.10
(1,2) → (1,1) + (1,1) 0.21
(0,0) → (1,0) + (1,0) 0.80
(0,1) → (1,1) + (1,1) 0.40
(0,0) → (1,1) + (1,1) 0.64
(0,1) → (1,0) + (1,1) 0.64
(0,1) → (1,1) + (1,0) 0.64
(0,2) → (1,0) + (1,0) 0.40
(0,2) → (1,1) + (1,1) 0.34
(1,0) → (0,0) + (1,0) 0.80
(1,1) → (0,1) + (1,1) 0.4
(1,0) → (0,1) + (1,1) 0.64
(1,1) → (0,0) + (1,1) 0.64
(1,1) → (0,1) + (1,0) 0.64
(1,2) → (0,0) + (1,0) 0.40
(1,2) → (0,1) + (1,1) 0.34
(1,0) → (1,0) + (0,0) 0.80
(1,1) → (1,1) + (0,1) 0.40
(1,0) → (1,1) + (0,1) 0.64
(1,1) → (1,0) + (0,1) 0.64
(1,1) → (1,1) + (0,0) 0.64
(1,2) → (1,0) + (0,0) 0.40
(1,2) → (1,1) + (0,1) 0.34
. . . → . . . + . . . . . .
20
Table 3.2: Some mode coupling possibilities in 2D waveguides

3.2 Pure heralded single photons
3.2.1 Quantum networks and heralding
The fidelity of pure heralded quantum states is of critical importance to quantum
networks. As a source of single photons, a lot of attention has been paid to the
process of spontaneous parametric downconversion. Signal and idler exhibit strict
photon number correlations, which can be utilised to herald the existence of the
signal photon by detecting the corresponding idler photon, resulting in a heralded
single photon source.
One easy way to separate the two-photon-state, after it left the waveguide, is
by the use of type-II downconversion and a polarizing beam splitter (PBS) (Figure
3.23). Detecting the idler photon to herald the presence of the signal photon is,
Figure 3.23: Heralding single photons with type-II PDC
in this case, equivalent to discarding all information about the idler photon. This
measurement process is modelled as a partial trace over the Hilbert space belonging
to the idler wave packet.
T ri |ψs,i〉 〈ψs,i| = ρs
(3.38)
After this detection the signal photon, in general, will not be in a pure single photon
state, but in a mixed single photon state ρs. If and only if the two-photon-state
|ψs,i〉 can be written as a product state |ψs〉 ⊗ |ψi〉, the heralded single photon will
be in a pure state.
T ri (|ψs〉 〈ψs| ⊗ |ψi〉 〈ψi|) = |ψs〉 〈ψs| ⊗ T ri |ψi〉 〈ψi| = |ψs〉 〈ψs| (3.39)
Two-photon-states that can be decomposed into a product state are called decorrelated
two-photon-states. Under what conditions is it possible to generate these
signal and idler photons into a product state? From the two-photon-state deduced
in Equation 3.23 we can immediately see that signal and idler states have to be
separable:
� ∞ � ∞
|ψs,i〉 = A
0 0 � ∞
!
= A
0
dωs dωif(ωs, ωi)â † sâ †
i |vac〉
dωs fs(ωs)â † s |0〉 ⊗
� ∞
0
dωi fi(ωi)â †
i |0〉 . (3.40)
21

Hence, we have to ensure a separable JSA as written in Equation 3.41.
f(ωs, ωi) = fs(ωs)fi(ωi) (3.41)
This condition can be depicted geometrically. Basically, the JSA is a two-dimensional
Gaussian distribution in frequency space.
f(x, y) = Ae −a (x−x0 )2
2σ2 x
−b (y−y0 )2
2σ2 −c
y
(x−x0 )(y−y0 )
2σxσy (3.42)
This distribution is separable, if and only if the parameter c equals zero. In that
case we obtain a two dimensional ellipse orientated along the coordinate axis, or a
case with a circular shaped JSA. That way it is possible to distinguish separable
and non-separable joint spectral amplitudes by a simple plot (see Figure 3.24).
Figure 3.24: Different two-photon states
To gain more insight into this problem we simplify the two-photon-state:
� ∞ � ∞
|ψs,i〉 = A dωs dωie − (ωp−ωs−ωi) 2
2σ2 L∆k
e
−γ( 2 ) 2
â † s(ωs)â †
i (ωi) |0〉 . (3.43)
0
0
We perform a Taylor series expansion up to the first order with slight detunings νs,i
around (ω0s, ω0i) [21].
ωs = ω0s + νs ωi = ω0i + νi ωp = ω0s + ω0i (3.44)
The momentum mismatch is expanded up to the first order and yields:
where
22
∆k(ωs, ωi) = kp(ωs + ωi) − ks(ωs) − ki(ωi) − 2π
Λ
⇒ ∆k(νs, νi) = ∆k0 + τsνs + τiνi, (3.45)
∆k0 = kp(ω0s + ω0i) − ks(ω0s) − ki(ω0i) − 2π
Λ
= 0,
τs = (k ′ p(ω0s + ω0i) − k ′ s(ω0s)), (3.46)
τi = (k ′ p(ω0s + ω0i) − k ′ i(ω0i)). (3.47)

This Taylor series expansion is performed at a point with perfect phasematching
(∆k0 = 0). Inserting Equations 3.44 to 3.45 into Formula 3.43 evaluates to:
� ∞ � ∞
|ψs,i〉 = A
0 0 � ∞ � ∞
= A
0
0
dωs dωie − (νs+ν i) 2
2σ 2 e
γL2
− 4
(τsνs+τiνi) 2
â † s(ωs)â †
i (ωi) |0〉
1
−
dωs dωie 2σ2 (ν2 s +ν2 γL2
i +2νsνi)− 4 (τ 2 s ν2 s +τ 2 i ν2 i +2τsτiνsνi) †
â s(ωs)â †
i (ωi) |0〉 .
From this equation, we obtain the factorizability condition [21]:
2
σ 2 + γL2 τsτi = 0,
(3.48)
2
σ 2 + γL2 (k ′ p − k ′ s)(k ′ p − k ′ i) = 0. (3.49)
The utilized crystal and laser have to meet this specific requirements to generate
uncorrelated photon pairs.
Further on with this formula, it is possible to determine the slope of ∆k = 0 for
every particular point of the contour in the {ωs, ωi}-plane (see Figure 3.25).
τsνs + τiνi = 0 (3.50)
� �
�
τs
k ′
p − k ′ �
s
⇒Θ = − arctan
τi
= − arctan
k ′ p − k ′ i
Figure 3.25: Definition of the phasematching angle
(3.51)
23

3.2.2 Frequency entanglement of two-photon-states
One possibility to quantify the spectral amount of entanglement is to perform a
Schmidt decomposition of the two-photon state. It is a transformation into a unique
set of orthonormal functions:
|ψs,i〉 = �
n
� λn |ψ n s 〉 ⊗ |ψ n i 〉 . (3.52)
The λn are the Schmidt coefficients and ψn s , ψn i are called Schmidt functions. With
the help of the λn it is possible to quantify the amount of entanglement. If only
one nonzero λn exists, we are coping with an uncorrelated two-photon-state. A high
number of Schmidt coefficients is equal to an entanglement in the frequency domain.
There are several possibilities to quantify the amount of entanglement. For example
the cooperativity parameter K, it equals 1 for an uncorrelated two-photon-state
and rises with successive contribution of Schmidt modes.
K =
1
� ∞
n=0 λ2 n
(3.53)
The entropy of entanglement is an alternative figure of merit and starts from 0
for an uncorrelated two-photon-state, also rising with increasing correlations.
S = −
∞�
λn log2(λn) (3.54)
n=0
The Schmidt decomposition, for a pure two-photon state, is performed by a numerical
singular value decomposition of the corresponding JSA. To illustrate this we
performed Schmidt decompositions of a correlated and an uncorrelated two-photonstate.
The Schmidt numbers in Figure 3.27 belong to the JSA plotted in Figure 3.26.
This two-photon-state exhibits a vast amount of Schmidt numbers, slowly decaying
for higher Schmidt modes ψ n s , ψ n i
, and hence of high cooperativity K = 33.93 and
entropy of entanglement S = 5.32.
In the case of a mostly uncorrelated JSA as plotted in Figure 3.30, the first Schmidt
number approaches 1 and the two-photon state is almost perfectly decorrelated
(K=1.20 and S=0.68) (Figure 3.31). Most signal and idler photons are emitted in
the first pair of Schmidt modes (Figures 3.32 and 3.33).
24

Ωi
f�Ωs,Ωi�
Ωs
Figure 3.26: Correlated JSA Figure 3.27: Schmidt numbers
0.2
0.1
�0.1
2.35 2.40
Ωs�PHz�
Figure 3.28: Schmidt functions signal
Ωi
f�Ωs,Ωi�
Ωs
0.2
0.1
�0.1
�0.2
2.35 2.40
Ωi�PHz�
Figure 3.29: Schmidt functions idler
Figure 3.30: Uncorrelated JSA Figure 3.31: Schmidt numbers
Figure 3.32: Schmidt function signal Figure 3.33: Schmidt function idler
25

3.3 Generating pure heralded single photons
We investigate different possibilities to generate pure heralded single photon states
with waveguided PDC. All plots and Schmidt decompositions, in this chapter and
the whole thesis, have been performed with consideration of the sinc term in the
two-photon state (Formula 3.22). All pump widths given in this section describe the
width of the pump intensity. This serves the purpose to make this data comparable
to units used in the laboratory.
3.3.1 Spectral filtering
The oldest and most straightforward method to create spectrally decorrelated photon
pairs is the application of spectral filtering. Most nonlinear crystals exhibit a
phasematching contour with a negative slope of -45 ◦ . This frequency distribution
can be modified by processing the photons through spectral filtering. In the experiment
a narrow spectral filter is placed in the beam path of the heralding photons
(see Figure 3.34). Filtering the signal arm would have the same effect, but in this
case a heralding detection event would not necessarily herald a signal photon. Its
partner may have been filtered.
Figure 3.34: Heralding filtered photons for higher purity
For simplicity we model the filter as a Gaussian function in frequency space.
Mathematically speaking we insert an additional filter function into our two-photonstate
3.23 and get:
� ∞ � ∞
|Ψs,i〉 = A
0
0
dωs dωi e − (ωi−ωf )2
2σ2 f e − (ωs+ωi−ωp)2 2σ2 ∆kL
p −γ(
e 2 )2
â † s (ωs) â †
i (ωi) |0〉 .
(3.55)
The effect of the filter function can be easily understood by an actual example
plotted in Figure 3.35. In the left picture a JSA with huge correlations is plotted.
The filter term, visualized in the second picture as the horizontal function, discards
a large part of the JSA and shapes it into a less correlated form. The result is
plotted in the figure to the right. The drawback of this method is the introduction
26

Figure 3.35: Unfiltered JSA, JSA + filter function, filtered JSA
of heavy photon loss.
In order to quantify the performance of this method we choose a common KTP
waveguide chip. In KTP most of the nonlinear indices dij are vanishing [22], with
the exception of:
❏ d15 = 3.7 pm/V
❏ d24 = 1.9 pm/V
❏ d31 = 3.7 pm/V
❏ d32 = 2.2 pm/V
❏ d33 = 14.6 pm/V
. Only a few processes remain:
⎛
⎝
Px
Py
Pz
⎞ ⎛
0
⎠ = ɛ0 ⎝ 0
3.7
0
0
2.2
0
0
14.6
0
1.9
0
3.7
0
0
⎞
0
0⎠
·
0
pm
V ·
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜
⎜2EyEz
⎟
⎝2EzEx
⎠
2ExEy
⎛
(Ex) 2
(Ey) 2
(Ez) 2
Five different processes are available in KTP, the strongest is a type-I downconversion
process with z-polarized pump, signal and idler fields. However we have
to separate the signal and idler photons after they leave the crystal and therefore
must excite a type-II downconversion process, despite the smaller nonlinearities as
opposed to type-I downconversion.
We choose a waveguide chip with parameters depicted in Table 3.3. The chip is
periodically poled in such a way that it fulfills phasematching conditions at λs,i =
800 nm. KTP as a material is very common and crystals with these parameters are
often applied in experiments investigating entanglement properties.
⎞
27

Material: KTP
Polarizations: y → y + z
400 nm → 800 nm + 800 nm
Λ: 8.85 µm
m: 1
Pump FWHM: 1.5 nm
Dimensions: 4 µm x 4 µm x 3.5 mm
Table 3.3: Investigated crystal and pump parameters
We define the loss in brightness as
Iloss =
� ∞ � ∞
0
dωs dωi e −2 (ω i −ω f )2
2σ 2 f f(ωs, ωi) 2
�0 ∞ � ∞
dωs dωif(ωs, ωi) 2
0
0
(3.56)
and calculate for every filter width the corresponding cooperativity parameter in
addition to the intensity loss. Our results are plotted in Figure 3.36. This plot
characterizes the problem with the filtering approach. For a suitable amount of
decorrelation filters with a width below 1 nm have to be applied. The tradeoff
between decorrelation and intensity loss is huge, but can be compensated by a longer
measurement time.
1/K and Brightness
1
0.8
0.6
0.4
0.2
Filtering the JSA
0
0 2 4 6 8 10 12 14 16 18 20 22
Filter sigma [nm]
Brightness
1/K
Figure 3.36: Loss in brightness and gain in purity for different filter bandwidths
In this crystal higher order spatial modes lead to a minimal shift in the phasematching
function and the result is a vast array of JSAs that produce downconverted
28

photons (Figure 3.37). These different JSAs lead to an increased distinguishability of
the signal and idler photons. Narrow frequency filters have to be applied to discard
all photons from unwanted JSAs, in order to handle these modal effects.
Ωi�PHz�
2.5
2.4
2.3
f�Ωs, Ωi�
2.2
2.2 2.3 2.4 2.5
Ωs�PHz�
Figure 3.37: JSAs created by different spatial modes propagating in the
waveguide
This approach to produce pure single-photon states is universal. It can be applied
to any two-photon-state. However the feasibility of quantum networks also depends
on a bright single photon source, a feature that this approach doesn’t provide.
3.3.2 Group velocity matching
In Formula 3.51 we already hinted another possibility to generate the desired photon
states. A positive phasematching slope in addition to the strictly descending pump
distribution would yield a circular shaped JSA and hence decorrelated photon pairs.
This approach depends on the crystal properties i.e., the Sellmeier equations, and
exhibits the advantage of discarding all need for filtering. As mentioned in the
introduction, this proposal has already been successfully implemented in bulk KDP
[10, 11].
In this thesis we want to extend this proposal and investigate different common
nonlinear materials. We suggest a KTP waveguide chip as shown in Table 3.4. The
first change is the wavelength range. Instead of creating photon pairs around 800
nm, we move up to 1550 nm. It is noteworthy that for this chip we make use of the
m = −1 phasematching order (Figure 3.38), as opposed to the filtering setup where
we used the m = 1 phasematching order in the same material.
The plotted JSA is not completely circular or elliptically shaped along one of the
coordinate axis. The reason for this behavior is that we also have to consider the
sinc contribution as stated in Formula 3.22. We plotted the same two-photon-state
29

Material: KTP
Polarizations: y → y + z
775 nm → 1550 nm + 1550 nm
Λ: 68.40 µm
m: -1
Pump FWHM: 0.88 nm
Dimensions: 4 µm x 4 µm x 5 mm
Table 3.4: Proposed and investigated crystal parameters
Figure 3.38: Pump envelope x phasematching function = JSA of the
propsed PDC state
with the additional side peaks of the sinc in figure 3.39. These peaks lead to an
increased distinguishability and limit the minimal achievable amount of entanglement.
The minimum cooperativity parameter possible with this proposed setup is
K=1.20 (S=0.67). It is important to note that a precise control over the pump
width is crucial because already detunings of about 0.5 nm result in a considerable
increase of the frequency correlations (see Figure 3.40). Effects from spatial modes
pose no problem. The phasematching curves, for different spatial modes, are too
much apart as plotted in Figure 3.41, and could be filtered with ease. Additionally
it is unlikely for wavelengths around 1550 nm to propagate in higher spatial modes
in a waveguide of 4 µm x 4 µm. In the experiment we observed only single mode
propagations in this wavelength regime.
Compared with the already implemented group velocity matching in KDP [10, 11],
this proposal exhibits two main advantages. The use of waveguides will result in
a much brighter single photon source and the wavelength of 1550 nm is optimal to
distribute the heralded photons through optical fibers.
30

Figure 3.39: Pump envelope x phasematching function = JSA with consideration
of the sinc contributions
Cooperativity parameter
2
1.8
1.6
1.4
1.2
Decorrelation
1
0 0.5 1 1.5 2 2.5 3
Pump sigma [nm]
Figure 3.40: Change of the cooperativity parameter by tuning of the
pump width
31

32
Ωi�PHz�
1.26
1.23
1.2
Φ�Ωs,Ωi�
1.17
1.17 1.2 1.23 1.26
Ωs�PHz�
Figure 3.41: Different phasematching functions due to spatial modes in
the proposed waveguides

3.3.3 Counterpropagating signal and idler fields
Until now we only considered generating decorrelated two-photon-states with strictly
copropagating signal and idler fields. In bulk crystal on the other hand, the use of
angular phasematching is a common and widespread method.
Counterpropagating photon pairs in waveguides have already been proposed in
2002 [23]. In this scenario, the created photon pairs are separated inside the crystal
and leave the waveguide at opposing ends (see Figure 3.42).
Figure 3.42: Counterpropagation in waveguided parametric downconversion,
energy conservation and momentum conservation
Describing the impact on the spectral correlations is straightforward. The pump
envelope α(ωs, ωi) remains unaltered, whereas in the phasematching function the
momentum mismatch is altered (Equation 3.31) as follows:
∆k(ωs, ωi) = kp(ωs + ωi) − ks(ωs)+ki(ωi) − 2πm
. (3.57)
Λ
The momentum vectors of signal and idler are opposite to each other. Thus the
crystal has to take a large amount of the pump momentum, hence very small grating
periods are needed.
Accordingly, the counterpropagation directly affects the decorrelation condition
3.49 which is changed to
2
σ 2 + γL2 (k ′ p − k ′ s)(k ′ p+k ′ i) = 0. (3.58)
This simple sign change has an immense effect on the photon creation. Its impact
can be understood by considering the impacts on the formula for the phasematching
angle, which is now given by:
Θ = − arctan
� k ′ p − k ′ s
k ′ p+k ′ i
�
. (3.59)
In most materials, the denominator will be much larger than the numerator. This
results in a phasematching contour oriented along the ωs-coordinate axis, independently
from the applied material. In addition, the sign change leads to a narrow
shaped contour.
33

The first appealing feature of counterpropagation field modes is the inherent separation
of the created signal and idler photons. This is in contrast to the copropagating
case, where the downconverted fields are created in identical spatial modes
and consequently have to be separated by the use of different polarizations . It is
now for the first time conceivable to herald photon pairs generated by type-I PDC
without resorting to a Sagnac-Loop [24].
Figure 3.43: Heralding counterpropagating photon pairs
The great advantage is that high χ (2) -susceptibilities are available in type-I PDC
processes. We investigated the properties of counterpropagating photon pairs in the
crystal with the largest nonlinearity available to date, PPLN:
❏ d31 = 4.64 pm/V
❏ d22 = 2.46 pm/V
❏ d33 = 32.4 pm/V
⎛
⎝
Px
Py
Pz
⎞ ⎛
0
⎠ = ɛ0 ⎝−2.46
4.64
0
2.46
4.64
0
0
32.4
0
1.9
0
0
0
0
⎞
−2.46
0 ⎠ ·
0
pm
V ·
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜
⎜2EyEz
⎟
⎝2EzEx
⎠
2ExEy
⎛
(Ex) 2
(Ey) 2
(Ez) 2
To generate pure heralded single photon states, we propose an experiment with
parameters shown in Table 3.5. As expected, this setup results in a very narrow
phasematching contour orientated along the signal axis (Figure 3.44).
A full Schmidt decomposition of this two-photon-state has been performed. The
eigenvalues λn are rapidly decreasing for increasing n (Figure 3.46). The achievable
amount of decorrelation is even better than in the previous copropagation approach
(K=1.08, S=0.26). Almost all weight is again on the first pair of Schmidt functions
(Figures 3.47 and 3.48).
As depicted in Figure 3.49, the decorrelation stays stable for small variations in
the pump width.
The different spatial modes (Figure 3.50), although generating different phasematching
contours near the main slope, do not introduce additional correlations.
34
⎞

Material: PPLN
Process : z → z + z
775 nm → 1550 nm + 1550 nm
Λ: 0.35 µm
m: 1
Pump FWHM: 0.21 nm
Dimensions: 4 µm x 4 µm x 5 mm
Table 3.5: Crystal parameters for a counterpropagating setup
Figure 3.44: Pump envelope, phasematching function and joint spectral
amplitude of the counterpropagating process
This is because the phasematching contours are orientated along the signal axis.
Additionally, most of these higher order spatial modes in the suggested waveguide
are unlikely to propagate, and therefore may not be observed at all.
In copropagating PDC the spectral distributions of signal and idler photons are
comparable, as opposed to counterpropagating-PDC where the spectral distribution
of the idler photon is more than one magnitude narrower than that of the signal
photon (see Figures 3.51 and 3.52). Again this is a result of the narrow phasematching
function orientated along the ωs-axis. This narrow spectrum may prove useful.
On one hand, the detector of the heralding photons may be optimized to match the
exact wavelength and hence operate on a high performance level. On the other hand,
this very narrow spectrum is optimal for the propagation through optical fibers, and
may be used to distribute quantum states over large distances.
The phasematching contour along the signal axis may also be used to create
a tuneable pure heralded single photon source. The heralding detector can stay
unchanged while the signal spectrum is changed to an arbitrary wavelength by a
simple tuning of the pump laser central frequency (Figure 3.53).
The feasibility of this photon source depends on the grating periods that can be
manufactured. To date the lowest grating period produced is Λ = 0.8µm in KTP
[25]. Hence, further technology progress is required to produce the needed Λ =
0.35µm. But owing to the importance of nonlinear materials substantial progress
during the next years is expected. One major drawback of counterpropagating PDC
35

Ωi�PHz�
0.10
0.05
1.22
1.215
1.21
f�Ωs,Ωi�
1.21 1.215
Ωs�PHz�
1.22
Figure 3.45: Uncorrelated JSA
1.212 1.214 1.216 1.218 1.220 Ωs�PHz�
Figure 3.47: Schmidt functions signal
0.1
0.05
Figure 3.46: Schmidt numbers
1.2148 1.2153 1.2158
Ωi�PHz�
Figure 3.48: Schmidt functions idler
is the reduced downconversion efficiency of about two orders of magnitude [26].
36

Cooperativity parameter
2
1.8
1.6
1.4
1.2
Decorrelation
1
0 0.1 0.2 0.3 0.4 0.5
Pump sigma [nm]
Figure 3.49: Cooperativity parameter over different pump widths
Figure 3.50: Different phasematching contours due to spatial modes in
waveguides
1.213 1.215 1.217
Ωs�PHz�
Figure 3.51: Signal spectrum
1.213 1.215 1.217
Figure 3.52: Idler spectrum
Ωi�PHz�
37

38
Figure 3.53: Different pump frequencies lead to a tuning of the signal
frequencies without altering the idler spectrum

3.3.4 Summary
We investigated three different approaches to generate pure heralded single photon
states in waveguides. Filtering the JSA is the oldest approach. It is already
implemented in the laboratory, yet suffers from a heavy loss in source brightness.
Group velocity matching discards all need for filtering and the simultaneous change
to work at the telecommunication wavelength will circumvent most problems with
spatial modes, in total resulting in a bright heralded single photon source. The
last approach, counterpropagation photon pairs, has exciting features such as a narrow
backpropagating spectrum, a tuneable copropagating spectrum and an inherent
signal-idler separation. Yet the manufacturing of the prerequisite short grating periods
is still a technological challenge.
39

4 Experiment
4.1 Avalanche photodiodes
We detect near infrared light at the single photon level. For this purpose, the most
suitable detectors available to date are avalanche photodetectors (APDs), based on
Indium Gallium Arsenide (InGaAs).
All APDs are, in principle, semiconductors with a pn-junction and an applied
reverse bias. Incoming photons are absorbed in the detector/semiconductor material
and create a free electron plus a corresponding hole. These two free charges are then
separated by the applied electric field. On their way through the detector material,
the charges are accelerated to a level, at which they create new electron-hole pairs.
This effect results in an electron/hole avalanche and generates a detectable current.
The most important parameters of single photon detectors are the quantum efficiency
and the noise or dark count level. Darks counts in this context are defined
as counts not related to the actual signal. They are generated from many different
sources, ranging from incoming stray light, clicks from thermal effects, PDC pump
light hitting the detector to afterpulsing. Afterpulses are detection events related to
remaining charge carriers from previous avalanches. These free charges may trigger
an avalanche without incoming photons.
The most common detector material is silicon. Silicon detectors are used to detect
photons in the range from 400 - 1000 nm. The large bandgap of silicon results in
a very low noise level. Most silicon detectors are operated at room temperature
in a free running mode i.e., they are always ready to detect photons. After each
avalanche, a deadtime is set. Any avalanches in this time bin are discarded. However,
silicon photodiodes are not able to detect light above 1µm. APDs based on
Germanium or InGaAs are much more efficient in this wavelength regime. Their
drawback is the higher dark count level, demand for active triggering and cooling.
4.2 id Quantique id201 APD
The id201 single photon detection modules are based on InGaAs. These detectors
are operated at -50 ◦ C to suppress noise from thermal effects. They are actively
triggered up to 8 MHZ with an applied narrow detection window in the nanosecond
regime, to reduce dark counts from stray light. After each avalanche, an adjustable
41

deadtime may be applied. This deadtime extends the time electrons and holes get
to recombine thus suppressing afterpulsing. The overall single photon detection
efficiency of these detectors can be set up to 25%, by adjusting the base voltage
(parameters in Figure 4.1, efficiency in Figure 4.2).
❏ Trigger Rate (0 - 8 MHz)
❏ Detector Pulse Width (2.5
ns, 5 ns, 20 ns, 50 ns, 100
ns)
❏ Detector Efficiency (10%,
15%, 20%, 25%)
❏ Detector Deadtime (None,
1 µs, 2 µs, 5µs, 10 µs)
Figure 4.1: id201 parameters Figure 4.2: id201 quantum efficiency
Before we used the APDs in an actual PDC experiment, we extensively tested
them. We investigated the dark count level, afterpulsing and additional noise from
remaining pump light guided into the detector.
4.2.1 Dark counts
First, we investigated the effects of stray light and thermal fluctuations. We monitored
the dark counts with a simple setup (see Figure 4.3). We blocked the input
to the detector, a single mode fiber for the near infrared, and applied an external
trigger source. The data acquisition for this experiment has been fully automated
and the trigger generator together with the APDs are remotely controlled (see Appendix
A.2). Our goal was to investigate the effects of trigger rates, pulse width,
detector efficiency and deadtime on the dark count rate. The results are visualised
in Figure 4.4 and 4.5 (for all measurement data, see Appendix 4.5).
As expected, the dark count rate rises linearly with the applied trigger frequency.
The critical parameter for the noise is the gating width. A doubling in gating
time from 2.5 ns to 5 ns (see Figure 4.4 and 4.5) leads to an increased count rate
over one order of magnitude. It is therefore crucial to apply the narrowest gating
possible. The nonlinear increase in the case of 5 ns gatings in Figure 4.5 is a result of
afterpulsing events. As also plotted in Figure 4.5 the afterpulsing can be significantly
reduced by deadtimes longer than 5 µs. For all data, please refer to Appendix A.3.1.
4.2.2 Afterpulsing
To further investigate the effect of afterpulsing, we used two cw-lasers centered
around 1310 nm and 1555 nm. These laser diodes where attenuated with a neutral
42

kcounts / second
0.14
0.12
0.1
0.08
0.06
0.04
0.02
Figure 4.3: Dark counts measurement setup
Gating width: 2.5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.4: Dark counts, 2.5 ns gating
kcounts / second
4
3.5
3
2.5
2
1.5
1
0.5
Gating width: 5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.5: Dark counts, 5 ns gating
density filter to simulate single photon sources. After the attenuation their light was
coupled into the detectors (see setup in Figure 4.6).
In this experiment incoming photons constantly hit the detector material. Therefore,
the trigger rate should be related to the count rate linearly. Any nonlinear
increase can be traced back to afterpulsing events. The measurement data plotted
in Figure 4.7 and 4.8 (all data in Appendix A.3.2 and A.3.3) shows that these assumption
is only valid in the case of low trigger rates. For trigger rates above 1
MHz, an additional deadtime is needed to restore the linear increase and discard
afterpulsing effects. This is very apparent in Figure 4.8. The count rate for no
applied deadtime (red graph) increases in a nonlinear manner above 1MHz. The
blue graph (1µs deadtime) exhibits a sharp drop at exactly 1 MHz trigger rate.
The reason for this behaviour is the deadtime of 1 µs. After each avalanche, the
deadtime overwrites the following gating pulse. After each detection the following
gating and hence the following possibility to detect an incoming photon is lost. As
a result, the effective detection efficiency of the APDs decreases. In this extended
43

Figure 4.6: Avalanche measurement setup
period between different detection events, free charge carriers have more time to
recombine and afterpulsing events are suppressed. The linear response is restored.
Nevertheless the deadtime has to be choosen with care. Too large deadtimes lead
to a saturation of the count rate (black graph in Figure 4.8). More detection events
than necessary are discarded.
kcounts / second
Gating width: 2.5 ns Detector efficiency 25%
30
25
20
15
10
5
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.7: 1330 nm laser detection events
4.2.3 Remaining pump light in PDC detection
kcounts / second
Gating width: 2.5 ns Detector efficiency 25%
800
700
600
500
400
300
200
100
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.8: 1550 nm laser detection events
According to the data sheet (Figure 4.2) the id201 APDs are not specified for any
light below 900 nm. We tested the APD response in this region, by the use of a 808
nm pulsed laser system. The results for different input powers and gating widths
are plotted in Figures 4.9 and 4.10 (all data in Appendix A.3.4).
Compared with the dark count rates in Figure 4.4, it is apparent that incoming
photons have been detected by the APDs. Either these counts are related to an
unspecified detection efficiency around 800 nm, or some fluorescence effects in the
fibres occurred. In all PDC experiments, the pump beam propagates in the same
beampath as the downconverted light, therefore it will be coupled into the fibres
44

kcounts per second
180
160
140
120
100
80
60
40
20
808nm pulsed laser 1nW, Width: 2.5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.9: 808 nm laser, 2.5 ns gating
kcounts per second
1800
1600
1400
1200
1000
800
600
400
200
808nm pulsed laser 1nW, Width: 5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Figure 4.10: 808 nm laser, 5 ns gating
leading to the detector. Actual PDC pump powers are in the mW regime but already
pump light over six magnitudes smaller than that introduces a significant level of
counts (figure 4.9 and 4.10). Evidently, pump filters with a suppression of 10 −6 or
higher have to be applied to cope with this dark count source. All measurement
data are again displayed in Appendix A.3.4.
4.2.4 Summary
Taking all different dark count sources into account, it is important to apply sufficient
deadtimes when the id201 is operated above 1MHz trigger rate. A narrow gating is
crucial to get rid of noise. Our laser systems produce pulses in the fs and ps regime
and enables us to use the narrowest gating of 2.5 ns. Also in the process of PDC it
is necessary to filter the remaining pump photons, as opposed to stray light, which
has no noticeable effect. All experiments can be done without the need for total
darkness in the laboratory.
45

4.3 Characterization of the waveguide chips
In this section, we give an overview of how we tested the phasematching characteristics
of our waveguide chips. All experiments served two main purposes. On the
on hand, we wanted to test how our theories compete with an actual experiment,
and on the other hand, we intended to identify the crystal most suitable to produce
degenerate and decorrelated photon pairs around 1550 nm. The final goal is to
implement the proposal discussed in Section 3.3.2.
We have been supplied with three different KTP crystals to perform experiments.
One unpoled waveguide chip served for testing and comparison purposes and two
crystals with a different step index poling period (Table 4.1). In all crystal chips
several waveguiding structures have been implemented as depicted in Figure 4.11.
Figure 4.11: Front view of the BCT0703-B12 chip (50x)
ChipID Grating Period Λ Waveguide length
ITI0706-B11 no poling 19 mm
BCT0703-B12 92.59 µm 11 mm
ITI0706-B12 104.17 µm 18 mm
Table 4.1: Different available waveguide chips
We tested the crystals in different SHG experiments. SHG generation has the
great advantage of huge conversion efficiencies, enabling us to work with standard
spectrometers and powermeters, unlike PDC, where the use of single photon detectors
is necessary.
4.3.1 Chip BCT0703-B12
Different SHG processes
At first, we turned to the question which processes can be excited in KTP. In KTP,
most of the nonlinear indices dij are vanishing [22]. In all three waveguide samples,
the fields are propagating in x-direction. That way, guided waves can only carry y-
46

or z-polarization, and we are left with three different processes:
Py = d242EyEz, (4.1)
Pz = d32E 2 y, (4.2)
Pz = d33E 2 z . (4.3)
Consequently, we should be able to identify three different conversion processes
(Ep1Ep2 → ESHG) [14]:
EyEy → Ez, (4.4)
EyEz → Ey, (4.5)
EzEz → Ez. (4.6)
A schematic representation of our setup can be seen in figure 4.12. As a pulsed
laser source, we used an optical parametric amplifier (OPA) that can generate pulsed
laser light from 1000 to 1700 nm with a FWHM around 15 nm. A small sample of
this beam has been split out to monitor the frequency, the main part has been
coupled into our waveguide. A half-wave plate in front of the crystal has been used
to change the polarisation of the incoming beam. After the crystal, we applied a
polarisation filter to test the polarization of the SHG light, whose frequency was
monitored by another spectrometer. The short-wave pass frequency filter, after the
waveguide, served the purpose to filter remaining pump light. Otherwise, the pump
light might damage the second spectrometer.
Figure 4.12: Setup to detect the different SHG processes
Into the waveguide, depicted in Figure 3.16, we sent z-polarised, y-polarised and
yz-polarized light. After the crystal we tested the SHG light for y- and z-polarisation.
This measurement has been performed multiple times at different pump wavelengths
ranging from 1200 to 1600 nm.
Our results for chip BCT0703-B12 can be seen in Figure 4.13 (all measurements
are listed in Appendix A.4.1). We where able to measure all SHG processes predicted
47

y theory. We detected a very strong type-I process (EzEz → Ez), a very weak
type-I process (EzEz → Ey), and one type-II process (EyEz → Ey). The process
EyEz → Ez is not a type-II downconversion, but the remainant of the strong type-I
PDC. Our future PDC setup producing separable two-photon states will employ the
observed type-II process. Surprisingly at all applied pump wavelengths the type-
II process is always much stronger than the weak type-I process in spite of their
comparable nonlinearities. This may be concering the change in polarisation in the
case of this specific type-I SHG, or may be traced back to the different phasematching
contours.
Figure 4.13: Different measured SHG processes in KTP
The accumulated data stresses the importance of precise control over the pump
polarization in future PDC-experiments. In type-II PDC, any misalignment in the
pump polarisation will introduce two type-I PDC processes that may be mistaken
for type-II downconversion and thus introduce noise.
Phasematching considerations
After identifying the different SHG processes, we restricted ourselves to the type-II
conversion process. To gain insight into its phasematching properties, we investigated
the mapping between the pump spectrum and the SHG spectrum produced
inside the crystal. The spectrum of a SHG pulse generated from a pulsed laser
source is given in Formula 3.7:
� ∞
I3(ω3) =
0
dω1K ′ I 2 pe − (ω 1 −ωp)2
2σ 2 p e − (ω 1 −ω 3 −ωp)2
2σ 2 p sinc 2
� �
∆kL
2
(4.7)
At a SHG far away of the momentum conservation condition, we can treat the
sinc factor as constant. Under these circumstances, we obtain a convolution of two
48

Gaussian functions. The result is a Gaussian distribution doubled in width and
doubled in central frequency:
I3(ω3) = K ′ e − (ω 3 −2ωp)2
4σ 2 p . (4.8)
With this formula, we can now map our measured pump data points into the expected
SHG spectrum by a simple transformation:
I(λp) → I
� �2 λp
, I(ωp) → I(2ωp)
2
2 . (4.9)
If we approach the phasematching, this formula is not valid any more. A numerical
simulation of this mapping without simplifications is displayed in Figure 4.14. As
plotted, the sinc function determines the conversion efficiency from pump to SHG
light.
The SHG spectrum gets boosted in magnitude near the phasematching point.
This effect will reveal the phasematching point ∆k = 0 when approached.
Figure 4.14: Mapping between SHG spectrum and pump spectrum as
predicted by theory
The setup for this experiment remained the same as the setup for the identification
of different SHG processes (see Figure 4.12). We monitored simultaneously the pump
and the SHG spectra, and choose type-II SHG generation. We pumped the crystal
with yz-polarized light and monitored the y-polarized SHG photons.
Different pump wavelengths between 1500 nm and 1600 nm were established. We
measured ’non-phasematched’ SHG as plotted in figure 4.15. In this Figure suitable
pump power creates only minimal SHG light. A great increase in SHG intensity
can be seen in Figure 4.16 around 780 nm. This reveals the phasematching point at
this wavelength. For this kind of measurement, it is necessary to guide enough laser
power inside the waveguide. Insufficient laser power will result in no detectable SHG
49

(Figure 4.17). The difference between phasematched and unphasematched SHG is
very instructive in Figure 4.18. Minimal pump power leads to SHG light at 780
nm. The conversion efficiency is much larger than at the peak to the left. For the
full data set please refer to Appendix A.4.2. Note that in Figures 4.15 to 4.18, the
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.15: Shifted SHG
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.17: Insufficient pump power
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.16: Phasematched SHG
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.18: Near phasematching
amplitudes of pump and SHG power have been scaled to illustrate the mappings
between SHG and pump light. Yet all plots in this measurement share the same
scaling factor.
Irregularities in the manufacturing process may lead to a different phasematching
contour for each different waveguide. Another measurement in a different waveguide
has been performed to investigate possible differences. This time no amplification
around 780 nm occurred as plotted in Figure 4.19. Furthermore at 800 nm a very
sharp peak was excited (Figure 4.20). This peak bears no resemblance to the prior
measurement and seems too narrow to result from phasematched SHG. One possible
explanation is remaining pump light from the optical amplifier that pumps the OPA.
This light around 800 nm may have seeded some SHG at this wavelength. Alternatively,
phasematched SHG may not have occurred because of insufficient pump
power. Given that we used another waveguide on the chip, it may also be possible
50

that the phasematching point was shifted out of the detection range. As an additional
remark the scaling is not comparable to the first measurements. Differences in
the incoupling into the spectrometers render any quantitative comparison between
the different measurement series impossible. For the full data set please refer to
Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.19: No phasematching
Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.20: New peak
appendix A.4.2.
To authenticate these measurements, we scanned another waveguide in the crystal,
from 1520 to 1660 nm pump wavelength (Figures 4.21 to 4.24). The data is in full
accordance with our first experiment. The spectral distributions are clearly amplified
towards the phasematching point at 780 nm, where a huge gain increase occurs (full
data in Appendix A.4.2).
Taking everything into account, we identified one phasematching point for crystal
BCT0703-B12 at 780 nm. At this point the chip fulfills the phasematching condition
and therefore it exhibits ∆k = 0.
How does the theory match with this data? To account for errors, we introduce a
new kE-vector into our phasematching function. This error or fitting vector is used
to fit the predictions of the theory to the experimental data.
∆k = kSHG − kp1 − kp2 − kΛ − kE
(4.10)
For crystal BCT0703-B12, with a poling period of 92 µm we get a kE-vector of
-0.02410 / µm (see Table 4.2).
kSHG 14.1787 / µm
kp1 6.97257 / µm
kp2 7.29812 / µm
kΛ 0.06786 / µm
-0.02410 / µm
kE
Table 4.2: Calculated k-vectors of the observed degenerate SHG process
51

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.21: No phasematching
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.23: Shifted SHG
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.22: Near phasematching
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Figure 4.24: Phasematching point
We can now use this error vector to correct the expected phasematching function
(see figure 4.25). In this graph we assumed ωp1 = ωp2 = ωp, and plotted the sinc
function of SHG, approximated as a Gaussian distribution.
φ(ωp) = e
L2
− (ky(ωSHG)−ky(ωp)−kz(ωp)−kΛ−kE) 2
4
(4.11)
The blue line shows the theoretically predicted phasematching point at 1300 nm,
the red line the new phasematching fitted to the experimental results.
The overall fitting constant kE is in the range of the grating vector. From
an expected phasematching point at 1300 nm, we had to adjust our model to a
phasematching at 1560 nm. Additionally, the theoretically predicted phasematching
ranges from 1550 nm to 1650 nm which is much broader than measured in the
experiment. This underlines the discrepancies between theory and experiment, and
may be solved by a better model of the waveguide geometry and hence more accurate
Sellmeier equations. The real phasematching contour will most probably exhibit a
steeper slope at the degeneracy point.
We scanned the two-dimensional phasematching plane of PDC on the +45 ◦ slope,
as plotted in Figure 4.26. The one dimensional plot in Figure 4.25 is a cut through
52

this two dimension plane on the black line. In total, we scanned the phasematching
plane on the 45 ◦ slope and detected the phasematching function at 1560 nm (black
circle).
Figure 4.25: Corrected phasematching Figure 4.26: Corrected phasematching 2D
4.3.2 Chip ITI0706-B12
In Chip ITI0706-B12 (Λ = 104.17µm) we omitted the search for different SHG
process and directly explored the phasematching properties. To investigate chip
ITI0706-B12, we developed a more robust method. We switched from spectral measurements
to power measurements. Our goal was to measure the intensity of the
SHG beam and pump laser simultaneously. At the phasematching point we should
measure a huge increase in the conversion efficiency, and therefore be able to detect
it with ease.
Gain measurement
The first important information for this method is the gain increase at different
pump powers. Neglecting losses, we define the SHG efficiency as:
SHGEfficiency =
ISHG
ISHG + IPump power after the crystal
. (4.12)
We constructed a setup (see Figure 4.27) where we monitored pump spectrum,
SHG spectrum, pump and SHG power after the waveguide simultaneously. The
pump light propagates through the long wave pass filter and is detected by a standard
powermeter. The polarizing beam splitter (PBS) reflects all type-I upconversion
light and all the type-II SHG light is measured in the second powermeter.
At three different pump wavelengths (figure 4.28) we measured the relationship
between pump and SHG powers, for different incident pump power levels.
As expected this process is nonlinear and the conversion efficiency increases with
rising pump power. This measurement shows the importance of a bright laser source.
53

Pump Power [a.u.]
25000
20000
15000
10000
5000
Figure 4.27: Measurement setup to simoultaneously detect SHG spectrum,
pump spectrum, SHG intensity and Pump intensity
λ p = 765 nm
λ p = 775 nm
λ p = 768 nm
Pump spectrum
0
740 750 760 770
λp [nm]
780 790 800
Figure 4.28: Applied pump spectra [nm/2]
SHG Power [µW]
7
6
5
4
3
2
1
λ p = 765 nm
λ p = 775 nm
λ p = 768 nm
SHG Gain
0
0 5 10 15 20 25 30 35
Input Power [µW]
Figure 4.29: Observed gain increase
Pump power levels below 5 µW are difficult to handle. Incoming SHG light remains
barely detectable and ambient light introduces additional noise. In this experiment
we detected a stronger SHG light at 765 nm than at 775 nm, at equal pump intensities.
This increase in conversion efficiency already predicts that the phasematching
point will be located below 1550 nm pump wavelength.
Comparison between different approaches
Before we switched to the alternate approach, we tested the equivalence of monitoring
the mapping between SHG and Pump spectrum and a measurement of the SHG
power. For this purpose we used the same setup as for the gain measurement 4.27.
The mappings plotted in Figures 4.30 to 4.33 show a phasematching around 760
nm SHG wavelength or at about 1520 nm pump wavelength. The full data is again
54

plotted in Appendix A.4.3.
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Figure 4.30: Unphasematched SHG
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Figure 4.32: Phasematched SHG
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Figure 4.31: Shifted SHG
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Figure 4.33: Shifted SHG
The result of the power measurements are plotted in Figure 4.34. For each different
pump wavelength, we measured the pump intensity after the waveguide and
the SHG power and calculated the SHG efficiency (Eq. 4.12). To eliminate any
nonlinear effects we always coupled the same amount of pump power through the
waveguide. The phasematching point resides at the peak of the Gaussian distribution.
A Gaussian fit reveals the phasematching point at exactly 1525 nm. This data
shows the equivalence of both approaches. However the measurement of conversion
efficiencies is superior. In the mapping approach we had to manually compare
the different spectra and examine all pictures for an increase in SHG. In the power
approach all data points measured are taken into account and the Gaussian fit to
the power distribution yields much more reliable and exact information about the
phasematching point.
Similar to the first crystal we used this data to fit our model against the experiment.
We had to introduce a kE of -0.01335/µm (Table 4.3), again a fitting vector
in the same order as the grating period. The corrected phasematching functions are
plotted in Figure 4.35 and 4.36, similar to the previous section. The black circle
55

56
SHG efficiency (a. u.)
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
SHG in the KTP waveguide with Λ = 104.17 µm
0
1440 1460 1480 1500 1520 1540 1560 1580
λp [nm]
Figure 4.34: Measured conversion efficiencies for different pump wavelengths

kSHG(762.5nm) 14.5506/ µm
kp1(1525nm) 7.15375 / µm
kp2(1525nm) 7.48836 / µm
kΛ=104.17µm 0.0603166 / µm
-0.01335 / µm
kerr
Table 4.3: Calculated k-vectors at the degeneracy point
indicates the phasematching peak we measured. Plot 4.35 shows that we should be
able to detect two additional phasematching points at 1629.18 nm where the m=-1
contour again crosses the +45 ◦ slope and at 1089.72 nm. Here the m=+1 contour
cuts the black line. Both points are indicated with green circles in the corresponding
Figures.
Figure 4.35: Corrected phasematching Figure 4.36: Corrected phasematching 2D
Further investigation
As a first step we simplified the setup. It is no longer necessary to monitor the
SHG spectra. We built up the setup drafted in Figure 4.37. The polarization
filter, as always, served to filter any type-I SHG processes. With two different
powermeters, we monitored the SHG power and the OPA power propagating through
the waveguide. Two frequency filters where applied to discard either the pump or
the SHG, in order to get a clean signal in the respective powermeter. The first
spectrometer before the waveguide was kept to monitor the OPA.
Our results can be seen in Figures 4.38 and 4.39 (all measurement data is presented
in Appendix A.4.3). Three phasematching points could be identified at 1138
nm, 1554 nm and 1627 nm, in the first measurement (Figure 4.39) and in the sec-
57

SHG Efficiency (a. u.)
1200
1000
800
600
400
200
Figure 4.37: Measurement setup to detect the pump intensity and SHG
intensity after the waveguide
SHG in the KTP waveguide Λ =104.17 mum, Pump Power = 15 µW
1400
0
1000 1100 1200 1300 1400 1500 1600 1700
λp [nm]
Figure 4.38: High power measurement
SHG efficiency (a. u.)
SHG in the KTP waveguide Λ = 104.17 mum, Pump Power = 8 µW
250
200
150
100
50
0
1400 1450 1500 1550
λp [nm]
1600 1650 1700
Figure 4.39: Low power measurement
ond measurement with a higher pump power, at 1542 and 1628 nm (Figure 4.38).
The differences in the detected phasematching points can be because of several reasons.
Between the first measurement with the peak at 1525 nm and the following
experiments we changed the waveguide in the crystal. Manufacturing errors most
likely have shifted the phasematching point, between the first measurement and the
subsequent ones in another waveguide. The slight difference in the peak positions
in figure 4.38 and 4.39, is most likely due to simple measurement uncertainties.
We adjusted the error vector to this new data, presented in figure 4.38 and got
Table 4.4. In Figures 4.40 and 4.41 we plotted the new corrected functions.
The results of this investigation are a successful detection of two different phasematching
orders (m = ±1, in crystal ITI0706-B12). Furthermore around 1600 nm
we could detect two degeneracy points for the same contour. Consequently between
this two points a positive slope with exactly +45 ◦ degrees exists and we will be able
to produce decorrelated photon pairs with this crystal.
58

kSHG(762.5nm) 14.2703/ µm
kp1(1525nm) 7.01723 / µm
kp2(1525nm) 7.34502/ µm
kΛ=104.17µm 0.0603166 / µm
-0.0313867 / µm
kerr
Table 4.4: Calculated k-vectors of the observed degenerate SHG process
Figure 4.40: Corrected phasematching Figure 4.41: Corrected phasematching 2d
The fit now relies on three data points, instead of one in the two previous section.
This is a great increase in reliability. Still with this one error parameter it is not
possible to move all three degeneracy points in the right places. In addition the kE
parameter is not invariant under the change of waveguides or grating periods. A
fully numerically approach to calculate the effective sellmeier equations may solve
the last discrepancies.
Nevertheless we gained important data. Despite minor quantitative variations
the prognoses are valuable, and as demonstrated in this chapter, have already been
tested successfully.
4.4 PDC experiments
Having evaluated the SHG experiments we decided to use crystal ITI0706-B12 with
a poling period of 104.17 µm. According to our data we should be able to generate
degenerate PDC photon pairs in the range between 1520 to 1560 nm.
The goal was to verify the existence of parametric downconversion in the predicted
frequency regime. To detect the photons we applied the id201 single photon counting
modules, tested in Section 4.1. Great care has to be taken not to confuse PDC with
59

dark counts, afterpulsing, stray light and remaining pump photons. To distinguish
all these effects we constructed a coincidence setup where we employed the strict
photon number correlation between signal and idler photons.
The setup sketched in Figure 4.42 is pumped at 760 nm with a pump FWHM
around 10 nm. We used a long wave cutoff filter of about 1000 nm and a short
wave cutoff filter at 700 nm to get rid of unwanted light from the laser system. Our
goal is to excite the type-II process in the crystal. Precise control over the pump
polarization is necessary therefore we controlled it with a Glan-Thompson polarizer.
The first half-wave plate in front of the polarizer served as a power control, the
second half-wave plate after the polarizer turned the pump polarization in the yplane.
Afterwards the beam is coupled into a polarisation maintaining fiber and fed
into the crystal. After the downconversion a long wave pass filter is applied to get
rid of the remaining pump light. Signal and idler beams are separated by a PBS,
coupled into single mode fibers and guided into the APDs. The signals have been
post processed with a time-to-digital converter (TDC), which evaluated the number
of coincidence counts [27].
Figure 4.42: Measurement setup to detect signal and idler photon-pairs
We achieved count rates around 3200 events per second for each detector and a
coincidence level of 94 coincidences per second (see table 4.5).
This setup may further be improved by the use of lenses to couple into the SMfibers.
At the moment we apply microscope objectives to couple into the single mode
fibers. The coupling efficiency is around 60%. The challenge is to couple the guided
modes emitted by the rectangular waveguide into the emission modes of a circular
single mode fiber. These two different modes don not perfectly match each other,
therefore the incoupling efficiency is limited and will never reach 100%.
60

Outlook
measurement signal counts/s idler counts/s coincidences/s
1 2829 3581 105
2 2947 3610 84
3 2873 3619 109
4 2861 3536 88
5 2847 3559 94
6 2851 3549 83
7 2893 3530 96
8 2813 3576 98
9 2908 3555 973
10 2773 3648 99
11 2878 3517 90
12 2876 3683 90
13 2774 3574 107
14 2858 3601 85
average 2855.79 3581.29 94.64
Table 4.5: PDC signal detection events, idler detection events and coincidence
counts per second
The next step would be to sample the JSA of the PDC with two monochromators in
front of the APDs, as depicted in Figure 4.43. The two monochromators select one
small part of the JSA (Figure 4.44) and the measured coincidence rate will herald
the value of f(ωs, ωi) at each sampled point. This will directly yield the created
JSA and test the presented SHG experiments.
The final step to a pure heralded single photon source is to verify the production
of decorrelated two-photon-states with a 4th-order Hong-Ou-Mandel dip [10].
61

Figure 4.43: Proposed JSA scanning setup Figure 4.44: Predicted JSA measurement
62

This is the silliest stuff that ever I heard.
William Shakespeare
A Midsummer Night’s dream
5 Conclusion and Outlook
In the past year considerable progress has been made towards a pure heralded single
photon source. We were able to extend the analysis of wave propagation inside
a waveguide to spatial modal effects in collaboration with Thomas Lauckner and
Division 3. Effects that have just recently been observed in experiments by Kaisa
Laiho.
We analyzed different possibilities to produce pure heralded single photon states
inside a waveguide ranging from the first attempts with spectral filtering to crystal
engineering, where uncorrelated photon pairs are produced directly in the downconversion
process. The first setups employing this method are currently under
development in the laboratory. We pushed this approach even further by investigating
the possibility of generating uncorrelated counterpropagating photon pairs
inside a waveguide geometry. The inherent decorrelation and separation of photon
pairs inside the crystal may be a further step to the integration and miniaturization
of these single photon sources. Several groups are currently facing the challenge of
producing chips with a submicrometer grating period.
Our characterization of the id201 APDs provided knowledge of their noise properties.
The gained data will prove useful for future work with these single photon
detectors.
SHG proved to be a valuable tool to investigate the phasematching properties of
our crystal samples. We were able to identify all degenerate phasematching points
and to compare our modelling with actual data. There are still discrepancies between
theory and experiment, but only moderate fitting is necessary to reach a good
agreement. Despite all simplifications in the derivation of the two-photon-state, we
showed that our theoretical treatment is sufficient. Further improvement could be
made by a more precise model of the waveguiding structures.
Finally we used the SHG data to find a PDC process degenerate in frequency. First
results are very promising. We are looking forward to see a bright pure heralded
single photon source in the near future.
63

A Appendix
A.1 Mathematica packages
For all calculations performed in this thesis we used Mathematica, where all the
needed formulas have been implemented into several packages. The main package
is called pdc.m, it covers the calculation of the pump envelope, the phasematching
function and the joint spectral amplitude, in all their different representations. An
assert function is provided by myAssert.m and myUnits.m handles the conversion
between different SI-units. The enhancedSellmeier package imports the refractive indices
for different nonlinear materials written in some text files previously calculated
with Matlab [20].
Useful formulas for the Schmidt decomposition are provided by the enhanced-
Sellmeier.m package. It has been written by Wolfram Helwig and can be traced
back to work of Wolfgang Maurer.
65

��
pdc package Ver: 1.0
Main package with useful functions for plotting an calculating PDC
Comments and Suggestions to: Andi
��
BeginPackage�"pdc‘",�"myUnits‘","myAssert‘","Notation‘","Units‘","enhancedSellmeier‘"��
Unprotect�pdchelp,getOpt,Λ,Ω,Fno,Fne,no,ne,nx,ny,nz,kwg,dk,dk2,findRootdk2,�,Α,Φgauss,Φsinc,jointSpecAm
,ΩplotjointSpecAmplitude
�
pdchelp::"usage"�"basic help file"
getOpt::"usage"�"getOpt"
Λ::"usage"�"Converts a given frequency Ω�Hertz� to the corresponding Λ�Meter�; Example: Λ�1.2
Ω::"usage"�"Converts a given wavelength Λ�Meter� to the corresponding frequency Ω�Hertz�; Example
Fno::"usage"� "Temperature Dependance of the refractive index in ppln.
Example: F�Temperature��30, tempScale��Units‘Celsius�
Default: Temperature��25, tempScale��Units‘Celsius"
Fne::"usage"� "Temperature Dependance of the refractive index in ppln.
Example: F�Temperature��30, tempScale��Units‘Celsius�
Default: Temperature��25, tempScale��Units‘Celsius"
no::"usage"�"refractive index for the ordinary ray in ppln.
Dependant on the wavelength�frequency of the incoming ray and the Temperature of the cristall
Example: no�Λ�� 1550 Nano Meter,Temperature��30, tempScale��Units‘Celsius�,
no�Ω�� 1.23 Peta Hertz,Temperature��30, tempScale��Units‘Celsius�
Defaults: Temperature��25, tempScale��Units‘Celsius"
ne::"usage"�"refractive index for the ordinary ray in ppln
Example: ne�Λ�� 1550 Nano Meter,Temperature��30, tempScale��Units‘Celsius�
ne�Ω�� 1.23 Peta Hertz,Temperature��30, tempScale��Units‘Celsius�
Defaults: Temperature��25, tempScale��Units‘Celsius"
nx::"usage"�"refractive index for the x�polarized ray in ktp
Example: nx��� 1550 Nano Meter�
nx��� 1.23 Peta Hertz�
Defaults: none"
ny::"usage"�"refractive index for the y�polarized ray in ktp
Example: ny��� 1550 Nano Meter�
ny��� 1.23 Peta Hertz�
Defaults: none"
nz::"usage"�"refractive index for the z�polarized ray in ktp
Example: nz��� 1550 Nano Meter�
nz��� 1.23 Peta Hertz�
Defaults: none"
noBBO::"usage"�"refractive index for the o�polarized ray in BBO
Example: nz��� 1550 Nano Meter�
nz��� 1.23 Peta Hertz�
Defaults: none"
neBBO::"usage"�"refractive index for the e�polarized ray in BBO
Example: nz��� 1550 Nano Meter�
nz��� 1.23 Peta Hertz�
Defaults: none"

2 pdc.m
kwg::"usage"�"Calculates the length of the k�vector in a waveguide
Example: kwg�sellmeier��no, �� 1550 Nano Meter, wgwidth�� 4 Micro Meter, Temperature�� 25, tempScale
kwg�sellmeier��nx, �� 1.23 Peta Hertz, wgwidth�� 4 Micro Meter�
Defaults: wgwidth�� 4 Micro Meter,Temperature��25, tempScale��Units‘Celsius"
dk::"usage"�"Calculates the k�vector mismatch for given signal, idler photons and grating in
Example: dk�np� no,ns� no,ni� no, Λs� 1550 Nano Meter, Λi � 1550 Nano Meter,�� 20 Micro Meter
dk�np� no,ns� no,ni� no, Ωs� 1.23 Peta Hertz, Ωi � 1.23 Peta Hertz�
Defaults: ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale��Units‘Celsius"
�::"usage"�"Calculates the phase mismatch for given signal, idler photons
Example: ��np� no,ns� no,ni� no, Λp� 775 Nano Meter, Λs� 1550 Nano Meter, Λi � 1550 Nano Meter
��np� no,ns� no,ni� no, Ωp� 2.46 Peta Hertz, Ωs� 1.23 Peta Hertz, Ωi � 1.23 Peta Hertz
Defaults: ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale��Units‘Celsius"
Α::"usage"�"Modelling of the pump as a Gaussian Puls
Example: Α�Λp� 775 Nano Meter, Λs� 1550 Nano Meter, Λi � 1550 Nano Meter, Λpwidth� 10 Nano Meter
Α�Ωp� 2.46 Peta Hertz, Ωs� 1.33 Peta Hertz, Ωi � 1.22 Peta Hertz , Ωpwidth� 0.01 Peta
Defaults: none"
Φgauss::"usage"�"Phasematching Function. Simplifyed to gaussian
Example: Φsinc�np� ne,ns� no,ni� no, Λs� 1550 Nano Meter, Λi � 1550 Nano Meter,�� �23.8 Micro
Φsinc�np� ne,ns� no,ni� no, Ωs� 1.23 Peta Hertz, Ωi � 1.23 Peta Hertz,�� �23.8 Micro
Defaults: wglength�� 5 Milli Meter ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale
Φsinc::"usage"�"Phasematching Function
Example: Φsinc�np� ne,ns� no,ni� no, Λs� 1550 Nano Meter, Λi � 1550 Nano Meter,�� �23.8 Micro
Φsinc�np� ne,ns� no,ni� no, Ωs� 1.23 Peta Hertz, Ωi � 1.23 Peta Hertz,�� �23.8 Micro
Defaults: wglength�� 5 Milli Meter ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale
jointSpecAmplitude::"usage"�"Spectral distribution function
Example: f�np� ne,ns� no,ni� no,Ωp� 2.45 Peta Hertz, Ωs� 1.24 Peta Hertz, Ωi � 1.24 Peta Hertz
f�np� ne,ns� no,ni� no,Λp� 775 Nano Meter, Λs� 1450 Nano Meter, Λi � 1450 Nano Meter
Defaults: wglength�� 5 Milli Meter ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale
jointSpecAmplitudeSinc::"usage"�"Spectral distribution function"
jointSpecIntensity::"usage"�"Spectral distribution function
Example: f�np� ne,ns� no,ni� no,Ωp� 2.45 Peta Hertz, Ωs� 1.24 Peta Hertz, Ωi � 1.24 Peta Hertz
f�np� ne,ns� no,ni� no,Λp� 775 Nano Meter, Λs� 1450 Nano Meter, Λi � 1450 Nano Meter
Defaults: wglength�� 5 Milli Meter ��� Infinity wgwidth�� 4 Micro MeterTemperature��25, tempScale
findRootdk2::"usage"�"Do not use."
dk2::"usage"�"Do not use"
ΛfilterSignal::"usage"�""
ΛfilterIdler::"usage"�""
ΩfilterSignal::"usage"�""
ΩfilterIdler::"usage"�""
Null

��
this is a list of all variables appearing in the package
variables with attribute "notSet" have to be declared,
while variables with default values can be neglected
The notSet is used in the If�� statements
��
Options�defaults���tempScale��Units‘Celsius,Temperature��25,
��"notSet",��"notSet",sellmeier��"notSet",
np��"notSet",ns��"notSet",ni��"notSet",
Λp��"notSet",Λs��"notSet",Λi��"notSet",
Ωp��"notSet",Ωs��"notSet",Ωi��"notSet",
Λpsigma��"notSet", Ωpsigma�� "notSet",
Λpfwhm��"notSet", Ωpfwhm�� "notSet",
��� Infinity Meter, ssig�� �1, isig�� �1,
wglength�� 5 Milli Meter,wgwidth�� Infinity,
Ωsrange��"notSet" ,Ωirange�� "notSet",
Λsrange�� "notSet",Λirange�� "notSet",
Ωsroot��"notSet",
Ωsmin��"notSet",Ωsmax��"notSet", Ωimin��"notSet", Ωimax��"notSet",
Λsmin��"notSet",Λsmax��"notSet", Λimin��"notSet", Λimax��"notSet",
optfoo��"notSet",optbar��"notSet",
modesup��0,modesvp��0,modesus��0,modesvs��0, modesui��0,modesvi��0,, modesu
Λsmin���500 Nano Meter�, Λimin���500 Nano Meter�, Λsmax�� �000 Nano Meter
kerror���0�Meter�,
ΛfilterSignalCenter�� "ΛfilterSignalCenter�notSet",ΛfilterSignalSigma�� "
ΛfilterIdlerCenter�� "ΛfilterIdlerCenter�notSet",ΛfilterIdlerSigma�� "ΛfilterIdlerS
ΩfilterSignalCenter�� "ΩfilterSignalCenter�notSet",ΩfilterSignalSigma�� "
ΩfilterIdlerCenter�� "ΩfilterIdlerCenter�notSet",ΩfilterIdlerSigma�� "ΩfilterIdlerS
�;
Begin�"‘Private‘"�
pdchelp�"
List of Implemented Functions. Type ?function for detailed instructions and examples.
"
�Fno temperature dependance of the sellmeier equations
�Fne
�no refractive index of the ordinary ray
�ne refractive index of the extra ordinary ray
�kwg k�vector in the waveguide
�dk k�vector mismatch
�� grating for given Ωs and Ωi
�Α pump
�Φgauss phasematching function with gauss
�Φsinc phasematching function with sinc
�jointSpecAmplitude joint�spectral amplitude without phase
�jointSpecIntensity joint�spectral intensity
The arguments can be placed in arbitrary order.
Some values if omitted are replaced by their default values. See function documentation
wglength:� length of the wabeguide
wgwidth:� width of the waveguide
��Speed of light��
c:��299792458.0 Meter��Second
��Assignment of the variables for use in the Modules��
getOpt�name�,opts���,func��:�Module���,Return�name�.�opts��.Options�func���;
pdc.m 3

4 pdc.m
���:�N�replaceSiPrefixes� �2 Рc���� �. Hertz�� 1�Second
���:�N�replaceSiPrefixes� �2 Рc���� �. Second �� 1�Hertz
��Begin: Sellmeier Equations for PPLN
Source:
Edward and Lawrence, Optical and Quantum Electronics16:373�4�1984�
D. Jundt, Optics Lettters Vol. 22 No.20 1555 �1997�
��
�� F�T� for the ordinary ray��
Fno�opts����:�Fno�opts��Module��optT,opttempScale,T0,optTcelsius,optF�,
T0�24.5;
optT�getOpt�Temperature,opts,defaults�;
opttempScale�getOpt�tempScale,opts,defaults�;
iassert�tempScale �� Celsius ��tempScale �� Kelvin �� tempScale �� Fahrenheit ��tempScale ��
optTcelsius�N�ConvertTemperature�optT,opttempScale,Celsius��;
iassert�optTcelsius�Reals�;
iassert�NumberQ�optTcelsius��;
iassert�optTcelsius���273.15�;
optF��optTcelsius�T0� �optTcelsius�T0�546�;
iassert�optF � Reals�;
iassert�NumberQ�optF��;
Return�replaceSiPrefixes�optF��;
�
��F�T� for the extraordinary ray��
Fne�opts����:�Fne�opts��Module��optT,opttempScale,T0,optTcelsius,optF�,
T0�24.5;
optT�getOpt�Temperature,opts,defaults�;
opttempScale�getOpt�tempScale,opts,defaults�;
iassert�tempScale �� Celsius ��tempScale �� Kelvin �� tempScale �� Fahrenheit ��tempScale ��
optTcelsius�N�ConvertTemperature�optT,opttempScale,Celsius��;
iassert�optTcelsius�Reals�;
iassert�NumberQ�optTcelsius��;
iassert�optTcelsius���273.15�;
optF��optTcelsius�T0� �optTcelsius�T0�546.32�;
iassert�optF�Reals�;
iassert�NumberQ�optF��;
Return�replaceSiPrefixes�optF��;
�

��refracrtive index of the o�polarized ray��
no�opts����:�no�opts��Module��optΛ,optn,optΩ, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optn:�Sqrt��4.9048��0.11775��2.2314 Fno�opts���10^8���optΛ^2��Micro^2 Meter^2���0.21802��2.9671
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��
�
��refracrtive index of the e�polarized ray��
ne�opts����:�ne�opts��Module��optΛ,optn,optΩ, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optn:�Sqrt��5.35583��4.629 Fne�opts���10^7��0.100473��3.862 Fne�opts���10^8���optΛ^2��Micro^2
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��
�
��End: Sellmeier Equations for ppln��
��Begin: Sellmeier Equations for ktp��
��
Source:
Waveguide Nonlinear�Optic Devices
T.Suhara M.Fujimura
S.310 Potassium Titanyl Phosphate�FTiOPO4
Sellmeier and tgerni.iotuc dusoersuib firnzkas fir KTP
Kiyoshi Kato and Eiko Takaoka
APPLIED OPTICS � Vol.41, No.24 � 20 August 2002
��
����2 are the old equations which are kept for safety reasons��
pdc.m 5

6 pdc.m
��refracrtive index of the x�polarized ray��
nx�opts����:�nx�opts��Module��optΛ,optn,optΩ, optneff, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optn:�Sqrt��3.29100 � 0.04140�� �optΛ^2��Micro^2 Meter^2���0.03978� � 9.35522� ��optΛ^2��Micro
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��
�
��refracrtive index of the y�polarized ray��
ny�opts����:�ny�opts��Module��optΛ,optn,optΩ, optneff, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optn:�Sqrt��3.45018 � 0.04341�� �optΛ^2��Micro^2 Meter^2���0.04597 � � 16.98825� ��optΛ^2��Micro
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��
�
��refracrtive index of the z�polarized ray��
nz�opts����:�nz�opts��Module��optΛ,optn,optΩ, optneff, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optn:�Sqrt��4.59423 � 0.06206�� �optΛ^2��Micro^2 Meter^2���0.04763 � � 110.80672� ��optΛ^2��Micro
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��
�
��End: Sellmeier Equations for ktp��

��Begin: Sellmeier Equations for BBO
Source:
Second�Harmonic Generation to 2048 A in beta�BaB2O4
K. Kato
IEEE journal of quantum electronics vol QE�22, NO.7 JULY 1986
��
��refracrtive index of the z�polarized ray��
noBBO�opts����:�noBBO�opts��Module��optΛ,optn,optΩ, optneff, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optneff:�Sqrt��2.7359 � 0.01878�� �optΛ^2��Micro^2 Meter^2���0.01822 � � 0.01354�optΛ^2��Micro
iassert�optneff��0�;
iassert�optneff�Reals�;
Return�replaceSiPrefixes�optneff��
�
��refracrtive index of the z�polarized ray��
neBBO�opts����:�neBBO�opts��Module��optΛ,optn,optΩ, optneff, optwgwidth�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�optΛ �� "notSet", optΛ :� Λ�optΩ��;
If�opt٠�� "notSet", opt٠:� �opt��;
iassert�optΛ �� 0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optneff:�Sqrt��2.3753 � 0.01224�� �optΛ^2��Micro^2 Meter^2���0.01667 � � 0.01516�optΛ^2��Micro
iassert�optneff��0�;
iassert�optneff�Reals�;
Return�replaceSiPrefixes�optneff��
�
��End: Sellmeier Equations for BBO��
pdc.m 7

8 pdc.m
��k�vector in the waveguide��
kwg�opts����:�kwg�opts��Module��optsellmeier,optwgwidth,optkwg,optΛ,optΩ�,
optΛ�getOpt�Λ,opts,defaults�;
��Print�"Λ: ",optΛ�;��
optΩ�getOpt�Ω,opts,defaults�;
��Print�"Ω: ",optΩ1�;��
iassert� Not� �optΛ �� "notSet"� && �optΩ �� "notSet"���;
If�opt٠�� "notSet", opt٠:� �opt� ,""�;
iassert�opt٠�� 0�;
iassert�optΩ Ε Reals�;
��Print�"optΩ: ", replaceSiPrefixes�optΩ��;��
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth Ε Reals�;
iassert�optwgwidth �� 0�;
optsellmeier�getOpt�sellmeier,opts,defaults�;
optkwg:��optsellmeier�opts� opt��c;
iassert�optkwg �� 0�;
iassert�optkwg Ε Reals�;
Return�Simplify�replaceSiPrefixes�optkwg���
�
��k�vector mismatch in a waveguide��
dk�opts����:�dk�opts��Module��optnp, optns, optni, optΛp, optΛs, optΛi,optΩp,optΩs,optΩi, optssig
optTemperature,opttempScale,optwgwidth, optwgheight, optairBoundary, optmodesup, optmodesvp,optmodesus
optTemperature�getOpt�Temperature,opts,defaults�;
opttempScale�getOpt�tempScale,opts,defaults�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optwgheight�getOpt�wgheight,opts,defaults�;
optpmorder�getOpt�pmorder,opts,defaults�;
optkerror�getOpt�kerror,opts,defaults�;
optdn�getOpt�dn,opts,defaults�;
optairBoundary�getOpt�airBoundary,opts,defaults�;
optmodesup�getOpt�modesup,opts,defaults�;
optmodesvp�getOpt�modesvp,opts,defaults�;
optmodesus�getOpt�modesus,opts,defaults�;
optmodesvs�getOpt�modesvs,opts,defaults�;
optmodesui�getOpt�modesui,opts,defaults�;
optmodesvi�getOpt�modesvi,opts,defaults�;
optssig�getOpt�ssig,opts,defaults�;
optisig�getOpt�isig,opts,defaults�;
iassert�optssig �� �1 �� optssig�� �1�;
iassert�optisig �� �1 �� optisig�� �1�;
opt��getOpt��,opts,defaults�;
iassert�opt� Ε Reals�;
optnp�getOpt�np,opts,defaults�;
;
;

optnp�getOpt�np,opts,defaults�;
optns�getOpt�ns,opts,defaults�;
optni�getOpt�ni,opts,defaults�;
iassert�optnp �� no �� optnp �� ne �� optnp �� nx �� optnp �� ny �� optnp �� nz �� optnp �� noe
iassert�optns �� no �� optns �� ne �� optns �� nx �� optns �� ny �� optns �� nz �� optns �� noe
iassert�optni �� no �� optni �� ne �� optni �� nx �� optni �� ny �� optni �� nz �� optni �� noe
optΛs�getOpt�Λs,opts,defaults�;
optΛi�getOpt�Λi,opts,defaults�;
optΩs�getOpt�Ωs,opts,defaults�;
optΩi�getOpt�Ωi,opts,defaults�;
If�optΩs �� "notSet", optΩs:�Ω�optΛs�;,""�;
If�optΩi �� "notSet", optΩi:�Ω�optΛi�;,""�;
iassert�optΩs Ε Reals�;
iassert�optΩs � 0�;
iassert�optΩi Ε Reals�;
iassert�optΩi � 0�;
optΩp � optΩs � optΩi;
optdk :� �kwg�sellmeier�� optnp,Ω�� optΩp,Temperature�� optTemperature, tempScale�� opttempScale
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesup, modesv
� optssig kwg�sellmeier�� optns,Ω�� optΩs,Temperature�� optTemperature, tempScale
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesus, modesv
� optisig kwg�sellmeier�� optni,Ω�� optΩi,Temperature�� optTemperature, tempScale
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesui, modesv
�2��optpmorder�opt��optkerror;
iassert�optdk Ε Reals�;
Return�Simplify�replaceSiPrefixes�optdk���;
�
��Calculation of the grating��
��opts����:���opts��Module��opt��,
opt�:� 2��dk�opts,���Infinity��;
iassert�opt� Ε Reals�;
Return�Simplify�replaceSiPrefixes�opt����;
�
��phasematching function��
Φgauss�opts����:�Φgauss�opts��Module��optΦgauss,optwglength�,
optwglength�getOpt�wglength,opts,defaults�;
iassert�optwglength �� 0�;
iassert�optwglength Ε Reals�;
optΦgauss :� �^��0.193 �0.5 optwglength dk�opts��^2�;
iassert�optΦgauss �� 0�;
iassert�optΦgauss Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΦgauss���;
�
pdc.m 9

10 pdc.m
��phasematching function��
Φsinc�opts����:�Φsinc�opts��Module��optΦsinc,optwglength�,
optwglength�getOpt�wglength,opts,defaults�;
iassert�optwglength �� 0�;
iassert�optwglength Ε Reals�;
optΦsinc :� Sinc�0.5 optwglength dk�opts��;
iassert�optΦsinc �� 0�;
iassert�optΦsinc Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΦsinc���;
�
��pump��
Α�opts����:�Α�opts��Module��optΑ,optΛp, optΛpsigma,optΛpfwhm, optΛs,optΛi, optΩp, optΩpsigma,
optΛp�getOpt�Λp,opts,defaults�;
optΛpsigma�getOpt�Λpsigma,opts,defaults�;
optΛpfwhm�getOpt�Λpfwhm,opts,defaults�;
optΛs�getOpt�Λs,opts,defaults�;
optΛi�getOpt�Λi,opts,defaults�;
optΩp�getOpt�Ωp,opts,defaults�;
optΩpsigma�getOpt�Ωpsigma,opts,defaults�;
optΩpfwhm�getOpt�Ωpfwhm,opts,defaults�;
optΩs�getOpt�Ωs,opts,defaults�;
optΩi�getOpt�Ωi,opts,defaults�;
If�optΛp �� "notSet", optΛp :� Λ�optΩp��;
iassert�optΛp �� 0�;
iassert�optΛp Ε Reals�;
If�optΛs �� "notSet", optΛs :� Λ�optΩs��;
iassert�optΛs �� 0�;
iassert�optΛs Ε Reals�;
If�optΛi �� "notSet", optΛi :� Λ�optΩi��;
iassert�optΛi �� 0�;
iassert�optΛi Ε Reals�;
If�optΛpsigma �� "notSet" && optΩpsigma �� "notSet" && optΩpfwhm �� "notSet", optΛpsigma � optΛpfwhm
If�optΛpsigma �� "notSet" && optΩpsigma �� "notSet" && optΛpfwhm �� "notSet", optΛpsigma � Λ�
If�optΛpsigma �� "notSet" && optΛpfwhm �� "notSet" && optΩpfwhm �� "notSet", optΛpsigma � Λ�optΩpsigma
iassert�optΛpsigma �� 0�;
iassert�optΛpsigma Ε Reals�;
optΑ:�Exp��1��optΛp��optΛs optΛi��optΛs�optΛi���^2��4�optΛpsigma�^2���;
iassert�optΑ �� 0�;
iassert�optΑ Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΑ���;
�
��joint spectral amplitude without phase��
jointSpecAmplitude�opts����:�jointSpecAmplitude�opts��Module��optjointSpecAmplitude�,
optjointSpecAmplitude:��Φgauss�opts� Α�opts��;
iassert�optjointSpecAmplitude Ε Reals�;
Return�Simplify�replaceSiPrefixes�optjointSpecAmplitude���;
�

��joint spectral amplitude without phase with sinc��
jointSpecAmplitudeSinc�opts����:�jointSpecAmplitudeSinc�opts��Module��optjointSpecAmplitudeSinc
optjointSpecAmplitudeSinc:��Φsinc�opts� Α�opts��;
iassert�optjointSpecAmplitudeSinc Ε Reals�;
Return�Simplify�replaceSiPrefixes�optjointSpecAmplitudeSinc���;
�
��joint spectral intensitiy��
jointSpecIntensity�opts����:�jointSpecIntensity�opts��Module��optjointSpecIntensity�,
optjointSpecIntensity:� �Α�opts� Φgauss�opts��^2;
iassert�optjointSpecIntensity �� 0�;
iassert�optjointSpecIntensity Ε Reals�;
Return�Simplify�replaceSiPrefixes�optjointSpecIntensity���;
�
pdc.m 11

12 pdc.m
��k�vector mismatch in a waveguide��
dk2�opts����:�dk2�opts��Module��optnp, optns, optni, optΛp, optΛs, optΛi,optΩp,optΩs,optΩi, optssig
optTemperature,opttempScale,optwgwidth, optwgheight, optairBoundary, optmodesu, optmodesv, optdn
optTemperature�getOpt�Temperature,opts,defaults�;
opttempScale�getOpt�tempScale,opts,defaults�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
optwgheight�getOpt�wgheight,opts,defaults�;
optpmorder�getOpt�pmorder,opts,defaults�;
optkerror�getOpt�kerror,opts,defaults�;
optdn�getOpt�dn,opts,defaults�;
optairBoundary�getOpt�airBoundary,opts,defaults�;
optmodesu�getOpt�modesu,opts,defaults�;
optmodesv�getOpt�modesv,opts,defaults�;
optΩsroot�getOpt�Ωsroot,opts,defaults�;
iassert�optΩsroot Ε Reals�;
optΛp�getOpt�Λp,opts,defaults�;
optΩp�getOpt�Ωp,opts,defaults�;
If�optΩp �� "notSet", optΩp:�Ω�optΛp�;,""�;
optssig�getOpt�ssig,opts,defaults�;
optisig�getOpt�isig,opts,defaults�;
iassert�optssig �� �1 �� optssig�� �1�;
iassert�optisig �� �1 �� optisig�� �1�;
opt��getOpt��,opts,defaults�;
iassert�opt� Ε Reals�;
optnp�getOpt�np,opts,defaults�;
optns�getOpt�ns,opts,defaults�;
optni�getOpt�ni,opts,defaults�;
iassert�optnp �� no �� optnp �� ne �� optnp �� nx �� optnp �� ny �� optnp �� nz �� optnp �� noe
iassert�optns �� no �� optns �� ne �� optns �� nx �� optns �� ny �� optns �� nz �� optns �� noe
iassert�optni �� no �� optni �� ne �� optni �� nx �� optni �� ny �� optni �� nz �� optni �� noe
optdk :� �kwg�sellmeier�� optnp,Ω�� optΩp,Temperature�� optTemperature, tempScale�� opttempScale
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesu, modesv
� optssig kwg�sellmeier�� optns,Ω�� optΩsroot,Temperature�� optTemperature, tempScale
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesu, modesv
� optisig kwg�sellmeier�� optni,Ω�� �optΩp �optΩsroot�,Temperature�� optTemperature
wgheight�� optwgheight, airBoundary�� optairBoundary, modesu �� optmodesu, modesv
�2��optpmorder�opt��optkerror;
iassert�optdk Ε Reals�;
Return�Simplify�replaceSiPrefixes�optdk���;
�

findRootdk2�opts����:�findRootdk2�opts��Module��optroot,optrootΩs,optΛs,optΩs�,
optroot�FindRoot�Meter�dk2�opts, Ωsroot�� root Peta Hertz�,�root,1��;
optrootΩs � root �. optroot;
Return�optrootΩs�;
�
�� Filter Functions ��
ΛfilterSignal�opts����:�ΛfilterSignal�opts��Module��optΛs, optΛfilterSignalCenter,optΛfilterSignalSigma
optΛs�getOpt�Λs,opts,defaults�;
optΛfilterSignalCenter�getOpt�ΛfilterSignalCenter,opts,defaults�;
optΛfilterSignalSigma�getOpt�ΛfilterSignalSigma,opts,defaults�;
optΛfilterSignal� Exp���optΛs�optΛfilterSignalCenter�^2��2 optΛfilterSignalSigma^2��;
iassert�optΛfilterSignal �� 0�;
iassert�optΛfilterSignal Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΛfilterSignal���;
�
ΛfilterIdler�opts����:�ΛfilterIdler�opts��Module��optΛi, optΛfilterIdlerCenter,optΛfilterIdlerSigma
optΛi�getOpt�Λi,opts,defaults�;
optΛfilterIdlerCenter�getOpt�ΛfilterIdlerCenter,opts,defaults�;
optΛfilterIdlerSigma�getOpt�ΛfilterIdlerSigma,opts,defaults�;
optΛfilterIdler� Exp���optΛi�optΛfilterIdlerCenter�^2��2 optΛfilterIdlerSigma^2��;
iassert�optΛfilterIdler �� 0�;
iassert�optΛfilterIdler Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΛfilterIdler���;
�
ΩfilterSignal�opts����:�ΩfilterSignal�opts��Module��optΩs, optΩfilterSignalCenter,optΩfilterSignalSigma
optΩs�getOpt�Ωs,opts,defaults�;
optΩfilterSignalCenter�getOpt�ΩfilterSignalCenter,opts,defaults�;
optΩfilterSignalSigma�getOpt�ΩfilterSignalSigma,opts,defaults�;
optΩfilterSignal� Exp���optΩs�optΩfilterSignalCenter�^2��2 optΩfilterSignalSigma^2��;
iassert�optΩfilterSignal �� 0�;
iassert�optΩfilterSignal Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΩfilterSignal���;
�
pdc.m 13

14 pdc.m
ΩfilterIdler�opts����:�ΩfilterIdler�opts��Module��optΩi, optΩfilterIdlerCenter,optΩfilterIdlerSigma
optΩi�getOpt�Ωi,opts,defaults�;
optΩfilterIdlerCenter�getOpt�ΩfilterIdlerCenter,opts,defaults�;
optΩfilterIdlerSigma�getOpt�ΩfilterIdlerSigma,opts,defaults�;
optΩfilterIdler� Exp���optΩi�optΩfilterIdlerCenter�^2��2 optΩfilterIdlerSigma^2��;
iassert�optΩfilterIdler �� 0�;
iassert�optΩfilterIdler Ε Reals�;
Return�Simplify�replaceSiPrefixes�optΩfilterIdler���;
�
End��;
EndPackage��;
Null

��self written assert package��
BeginPackage�"myAssert‘"�
��iassert is used for internal asserts in the pdc package��
��assert should be used for external asserts in the notebook��
assert::usage�"assert �condition� aborts if condition is false.
Use for assert in extern Functions"
iassert::usage�"assert �condition� aborts if condition is false.
Use for assert in intern Functions"
Begin�"‘Private‘"�
assert�condition�� :� If�condition,"",Abort���
iassert�condition�� :� If�condition,"",Abort���
End��
EndPackage��

��self written units package��
��Enhancement of the Mathematica Units‘ Package��
��Conversion of the SiPrefixes to actual Numbers��
BeginPackage�"myUnits‘","Units‘"�
replaceSiPrefixes::usage�"Replaces all SI�Prefixes with their actual value i.e. Micro �� 10^�
placeSiPrefixes::usage�"Replaces all Exponents with their actual SI�Prefixes i.e. 10^�3 �� Micro
si::usage�"make nice output"
$Assumptions :� � Meter � 0, Meter Ε Reals,
Second � 0, Second Ε Reals,
Hertz � 0, Hertz Ε Reals,
Celsius �0, Celsius Ε Reals�
Begin�"‘Private‘"�
replaceSiPrefixes�function�� :�
function �. �
Yocto �� 10^�24,
Zepto �� 10^�21,
Atto �� 10^�18,
Femto �� 10^�15,
Pico �� 10^�12,
Nano �� 10^�9,
Micro �� 10^�6,
Milli �� 10^�3,
Centi �� 10^�2,
Deci �� 10^�1,
�
Deca �� 10^1,
Hecto �� 10^2,
Kilo �� 10^3,
Mega �� 10^6,
Giga �� 10^9,
Tera �� 10^12,
Peta �� 10^15,
Exa �� 10^18,
Zetta �� 10^21,
Yotta �� 10^24,
Hertz �� 1�Second
placeSiPrefixes�function�� :�
N�function �. �
10^�24 �� Yocto,
10^�23 �� 10^�2 Zepto,
10^�22 �� 10^�1 Zepto,
10^�21 �� Zepto,
10^�20 �� 10^�2 Atto,
10^�19 �� 10^�1 Atto,
10^�18 �� Atto,
10^�17 �� 10^�2 Femto,
10^�16 �� 10^�1 Femto,
10^�15 �� Femto,
10^�14 �� 10^�2 Pico,
10^�13 �� 10^�1 Pico,
10^�12 �� Pico,
10^�11 �� 10^�2 Nano,
,

2 myUnits.m
��
10^�11 �� 10^�2 Nano,
10^�10 �� 10^�1 Nano,
10^�9 �� Nano,
10^�8 �� 10^�2 Micro,
10^�7 �� 10^�1 Micro,
10^�6 �� Micro,
10^�5 �� 10^�2 Milli,
10^�4 �� 10^�1 Milli,
10^�3 �� Milli,
10^�2�� Centi,
10^�1 �� Deci,
10^1 �� Deca,
10^2 �� Hecto,
10^3 �� Kilo,
10^4 �� 10^1 Kilo,
10^5 �� 10^2 Kilo,
10^6 �� Mega,
10^7 �� 10^1 Mega,
10^8 �� 10^2 Mega,
10^9 �� Giga,
10^10 �� 10^1 Giga,
10^11 �� 10^2 Giga,
10^12 �� Tera,
10^13 �� 10^1 Tera,
10^14 �� 10^2 Tera,
10^15 �� Peta,
10^16 �� 10^1 Peta,
10^17 �� 10^2 Peta,
10^18 �� Exa,
10^19 �� 10^1 Exa,
10^20 �� 10^2 Exa,
10^21 �� Zetta,
10^22 �� 10^1 Zetta,
10^23 �� 10^2 Zetta,
10^24 �� Yotta,
10^25 �� 10^1 Yotta,
10^26 �� 10^2 Yotta
si�function�� :� placeSiPrefixes�replaceSiPrefixes�function��;
End��
EndPackage��

��this packages imports the sellmeier equations calculated with matlab��
BeginPackage�"enhancedSellmeier‘",�"myUnits‘","myAssert‘","Notation‘","Units‘"��
noe::usage�"enhanced Sellmeier equation for PPLN."
nee::usage�"enhanced Sellmeier equation for PPLN."
nxe::usage�"enhanced Sellmeier equation for KTP."
nye::usage�"enhanced Sellmeier equation for KTP."
nze::usage�"enhanced Sellmeier equation for KTP."
nxe2::usage�"enhanced Sellmeier equation for KTP."
nye2::usage�"enhanced Sellmeier equation for KTP."
nze2::usage�"enhanced Sellmeier equation for KTP."
��this is a list of all variables appearing in the package
variables with attribute "notSet" have to be declared,
while variables with default values can be neglected
The notSet is used in the If�� statements��
Options�defaults���tempScale��Units‘Celsius,Temperature��25,
��"notSet",��"notSet",sellmeier��"notSet",
np��"notSet",ns��"notSet",ni��"notSet",
Λp��"notSet",Λs��"notSet",Λi��"notSet",
Ωp��"notSet",Ωs��"notSet",Ωi��"notSet",
Λpsigma��"notSet",Ωpsigma��"notSet",Λpfwhm��"notSet",Ωpfwhm��"notSet",
���Infinity Meter,ssig���1,isig���1,
wglength��5 Milli Meter,wgwidth��4 Micro Meter,
Ωsrange��"notSet",Ωirange��"notSet",Λsrange��"notSet",Λirange��"notSet",
Ωsroot��"notSet",
Ωsmin��"notSet",Ωsmax��"notSet",Ωimin��"notSet",Ωimax��"notSet",
Λsmin��"notSet",Λsmax��"notSet",Λimin��"notSet",Λimax��"notSet",
optfoo��"notSet",optbar��"notSet",
modesu��0,modesv��0,airBoundary��"",dn��"notSet",wgheight��"notSet"
�;
Begin�"‘Private‘"�
��Assignment of the variables for use in the modules��
getOpt�name�,opts���,func��:�Module���,Return�name�.�opts��.Options�func���;
��refractive index of the o�polarized ray in ppln��
noe�opts����:�noe�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
;

2 enhancedSellmeier.m
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "o" �� optairBoundary �� "e" ��optairBoundary �� ""�;
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu�getOpt�modesu,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv�getOpt�modesv,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "o",optdirectory � ".�neff�fit�ppln�true�ordinary�neff�fit�true�ordinary
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu� �� "�" �� ToString�optmodesv��� "�" ��
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
;

opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
��refractive index of the e�polarized ray in ppln��
nee�opts����:�nee�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "o" �� optairBoundary �� "e" ��optairBoundary �� ""�;
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu�getOpt�modesu,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv�getOpt�modesv,opts,defaults�;
iassert�optmodesv ��0�;
;
enhancedSellmeier.m 3

4 enhancedSellmeier.m
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "e",optdirectory � ".�neff�fit�ppln�true�extraordinary�neff�fit�true�extraordinary
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu� �� "�" �� ToString�optmodesv��� "�" ��
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�

��pump modes��
��refractive index of the x�polarized ray in ppktp ��
nxe�opts����:�nxe�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu�getOpt�modesu,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv�getOpt�modesv,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "x",optdirectory � ".�neff�fit�ktp�true�EX�neff�fit�true�EX�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu� �� "�" �� ToString�optmodesv��� "�" ��
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
;
enhancedSellmeier.m 5

6 enhancedSellmeier.m
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
��refractive index of the y �polarized ray in ppktp��
nye�opts����:�nye�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
;
;

optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu�getOpt�modesu,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv�getOpt�modesv,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "y",optdirectory � ".�neff�fit�ktp�true�EY�neff�fit�true�EY�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu� �� "�" �� ToString�optmodesv��� "�" ��
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
enhancedSellmeier.m 7
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;

8 enhancedSellmeier.m
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
��refractive index of the z�polarized ray in ppktp��
nze�opts����:�nze�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu�getOpt�modesu,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv�getOpt�modesv,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��

��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "z",optdirectory � ".�neff�fit�ktp�true�EZ�neff�fit�true�EZ�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu� �� "�" �� ToString�optmodesv��� "�" ��
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
�� Second set of enhanced sellmeier equations��
enhancedSellmeier.m 9

10 enhancedSellmeier.m
��refractive index of the x�polarized ray in ppktp ��
nxe2�opts����:�nxe2�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu2�getOpt�modesu2,opts,defaults�;
iassert�optmodesu2 ��0�;
iassert�optmodesu2 Ε Reals�;
Print�"optmodesu2: ",optmodesu2�;
optmodesv2�getOpt�modesv2,opts,defaults�;
iassert�optmodesv2 ��0�;
iassert�optmodesv2 Ε Reals�;
Print�"optmodesv2: ",optmodesv2�;
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "x",optdirectory � ".�neff�fit�ktp�true�EX�neff�fit�true�EX�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu2� �� "�" �� ToString�optmodesv2��� "�"
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
;

opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
��refractive index of the y �polarized ray in ppktp��
nye2�opts����:�nye2�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
enhancedSellmeier.m 11

12 enhancedSellmeier.m
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu2�getOpt�modesu2,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv2�getOpt�modesv2,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
If�optairBoundary �� "y",optdirectory � ".�neff�fit�ktp�true�EY�neff�fit�true�EY�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu2� �� "�" �� ToString�optmodesv2��� "�"
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
;

optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
��refractive index of the z�polarized ray in ppktp��
nze2�opts����:�nze2�opts��Module��optn,optΛ,optΩ, optwgwidth, optwgheight, optdn, optairBoundary
optstring, optdirectory, optpath, optcoefficient, optstringdn,optdnmantissa,optdnexponent
opta, optb, optc, optd,opte,optf, optg�,
optΛ�getOpt�Λ,opts,defaults�;
optΩ�getOpt�Ω,opts,defaults�;
iassert�Not��opt��"notSet"�&&�opt��"notSet"���;
If�optΛ��"notSet",optΛ:�Λ�optΩ�,""�;
optΛ � replaceSiPrefixes�optΛ��Micro Meter��;
iassert�opt��0�;
iassert�optΛ Ε Reals�;
optwgwidth�getOpt�wgwidth,opts,defaults�;
iassert�optwgwidth��0�;
iassert�optwgwidth Ε Reals�;
��Print�"optwgwidth: ",optwgwidth�;��
optwgheight�getOpt�wgheight,opts,defaults�;
iassert�optwgheight ��0�;
iassert�optwgheight Ε Reals�;
��Print�"optwgheight: ",optwgheight�;��
optdn�getOpt�dn,opts,defaults�;
iassert�optdn ��0�;
iassert�optdn Ε Reals�;
��Print�"optdn: ",optdn�;��
optairBoundary�getOpt�airBoundary,opts,defaults�;
iassert�optairBoundary �� "x" �� optairBoundary �� "y" �� optairBoundary �� "z" �� optairBoundary
��Print�"optairBoundary: ",optairBoundary�;��
optmodesu2�getOpt�modesu2,opts,defaults�;
iassert�optmodesu ��0�;
iassert�optmodesu Ε Reals�;
��Print�"optmodesu: ",optmodesu�;��
optmodesv2�getOpt�modesv2,opts,defaults�;
iassert�optmodesv ��0�;
iassert�optmodesv Ε Reals�;
��Print�"optmodesv: ",optmodesv�;��
��gruesome code to assemble the approbriate filename��
enhancedSellmeier.m 13

14 enhancedSellmeier.m
If�optairBoundary �� "z",optdirectory � ".�neff�fit�ktp�true�EZ�neff�fit�true�EZ�25�",optdirectory
optdnmantissa�PaddedForm�10 MantissaExponent�optdn���1��,�8,6��;
optdnexponent��MantissaExponent�optdn���2����1;
optstringdn � ToString�optdnmantissa� �� "e�0" �� ToString��1�optdnexponent��;
optwgwidth�PaddedForm�replaceSiPrefixes�optwgwidth��Micro Meter��,�8,6��;
optwgheight�PaddedForm�replaceSiPrefixes�optwgheight��Micro Meter��,�8,6��;
optpath � ToString�optdirectory� �� ToString�optmodesu2� �� "�" �� ToString�optmodesv2��� "�"
optpath � StringReplace�optpath,�" " �� ""��;
��Print�"optpath: ", optpath�;��
��extracts the coefficients from the selected file��
opta � FindList�ToString�optpath�,"a�"�;
opta � StringReplace�ToString�opta�,�"e�"��" 10^","e�"��" 10^�","a�" ��"", "�"�� "", "�"��""�
opta � ToExpression�opta�;
��Print�"opta: ", opta�;��
optb � FindList�ToString�optpath�,"b�"�;
optb � StringReplace�ToString�optb�,�"e�"��" 10^","e�"��" 10^�","b�" �� "", "�"�� "", "�"��""
optb � ToExpression�optb�;
��Print�"optb: ", optb�;��
optc � FindList�ToString�optpath�,"c�"�;
optc � StringReplace�ToString�optc�,�"e�"��" 10^","e�"��" 10^�","c�" �� "", "�"�� "", "�"��""
optc � ToExpression�optc�;
��Print�"optc: ", optc�;��
optd � FindList�ToString�optpath�,"d�"�;
optd � StringReplace�ToString�optd�,�"e�"��" 10^","e�"��" 10^�","d�" �� "", "�"�� "", "�"��""
optd � ToExpression�optd�;
��Print�"optd: ", optd�;��
opte � FindList�ToString�optpath�,"e�"�;
opte � StringReplace�ToString�opte�,�"e�"��" 10^","e�"��" 10^�","e�" �� "", "�"�� "", "�"��""
opte � ToExpression�opte�;
��Print�"opte: ", opte�;��
optf � FindList�ToString�optpath�,"f�"�;
optf � StringReplace�ToString�optf�,�"e�"��" 10^","e�"��" 10^�","f�" �� "", "�"�� "", "�"��""
optf � ToExpression�optf�;
��Print�"optf: ", optf�;��
optg � FindList�ToString�optpath�,"g�"�;
optg � StringReplace�ToString�optg�,�"e�"��" 10^","e�"��" 10^�","g�" �� "", "�"�� "", "�"��""
optg � ToExpression�optg�;
��Print�"optg: ", optg�;��
��calculation of n��
optn :� Sqrt�opta � optb��optΛ^2�optc�� optd optΛ��optΛ^2 � opte�� optf �optΛ�optg��;
iassert�optn��0�;
iassert�optn�Reals�;
Return�replaceSiPrefixes�optn��;
�
End��;
EndPackage��;

��package with useful functions for Schmidt decompositions with Mathematica��
BeginPackage�"wmschmidt‘"�
getOpt�name�,opts���,func��:�Module���,Return�name�.�opts��.Options�func���;
��ListIntegrate2D�data�List,�dx�,dy���:�ListIntegrate�Map�ListIntegrate��,dy�&,data�,dx�;��
convertDomain�doConv�,val�,iniDom�,destDom��:�convertDomain�doConv,val,iniDom,destDom��Return
�� Convert a discrete representation of a function back into an object which looks like a regular
function to Mathematica. x,y denote to coordinates, range defines the function domain as
�startX, endX, startY, endY�, steps � �stepsX, stepsY� gives the number of cells in each
grid direction, and data carries the array with the discrete function values. ��
convertMatrixToFunction�x�, y�, range�, steps�, data�� :�
Module��n,m, startX, endX, startY, endY, stepsX, stepsY�,
startX � range��1��;
endX � range��2��;
startY � range��3��;
endY� range��4��;
stepsX � steps��1��;
stepsY � steps��2��;
n � Round��x�startX����endX�startX���stepsX�1����1;
m � Round��y�startY����endY�startY���stepsY�1����1;
��Print�"Looking up �", n, ", ", m, "�"�; ��
Return�data��n,m���;
�;
�� Convert a function into a discrete representation. steps and range are defined as above, fn
is a funcion of two arguments, i.e., z � f�x,y�. ��
convertFunctionToMatrix�range�, steps�, fn�� :�convertFunctionToMatrix�range, steps, fn� �
Module��startX, endX, startY, endY, stepsX, stepsY�,
startX � range��1��;
endX � range��2��;
startY � range��3��;
endY� range��4��;
stepsX � steps��1��;
stepsY � steps��2��;
Return�Table�fn�startX � n��endX�startX��stepsX,startY � m��endY�startY��stepsY�, �n,1,stepsY
�;
�� A wrapper function which can be used where a function with signature z � func�x,y� is expected
convertM2FWrap�range�, steps�, data�� :�Function ��x,y�, convertMatrixToFunction�x,y,range,steps
�� Define some orthogonal polynomials: Hermite �normH�, Legendre�normL�, TODO: More ��
�� TODO: Use more reasonable function names ��
normH�n�, x�� :� Exp��x^2�2��1�Sqrt�2^n�n��Sqrt�Pi���HermiteH�n,x�;
normL�n�, x�� :� Sqrt��2n�1��2��LegendreP�n,x�;
�� Given a basis of a �separable� Hilbert space indexed by n, this procedure computes the overlap
� dx� dy Subscript�e^�, m��x�Subscript�e^�, m��y�psi�x,y�. By default, Hermite polynomials with
weight are used as basis functions on an infinite carrier. Setting the optional argument
basisFn allows to specify a different set of basis functions which are expected to be
of the from basis�m,x� where m is an integer argument denoting the m^th basis element and x is
continuous on the carrier. �All options described in the following are optional�.
If convertDomain is set to True, compact carriers for the function psi and the basis are used
Both carriers need not be identical. If, for instance, Legendre polynomials are used, the basis
domain will range from �1 to 1, i.e., basisDomain����1,1�. If a function defined on the
compact carrier �0,5� is supposed to be integrated, funcDomain���0,5� needs to be specified.

2 wmschmidt.m
compact carrier �0,5� is supposed to be integrated, funcDomain���0,5� needs to be specified.
This is independent of the range of integration which can be set using range���Subscript�x, min
If the main contributions of the abovementioned function are expected in the inner part of the
domain, it could for instance be reasonable to use range���1,4,1,4� to avoid integration over
voluminous vanishing contributions �which can lead to numerical instabilities�.
The accuracy goal for the integration can be set using accGoal. See the documentation on integrate
what this parameter is good for. This option does have no effect when discrete functions are
If discreteFunction is set to True, psi�x,y� needs to be specified as an array of discrete function
values. A ListIntegration is performed in this case since NIntegrate does unfortunately not work
gridded data. The grid step sizes are automatically determined from the dimensions of the discrete
and the integration range.
If showProgress is set to True, the routine gives information about which matrix element
is evaluated right now. TODO: Is there some mechanism similar to infolevel which could be used
overlap�m�, n�, psi�, opts���� :� overlap�m,n,psi,opts� � Module��optRange, optBasisFn, optShowProgress
optConvertDomain, optGridStepSize, valueTable�,
optRange � getOpt�range, opts, overlap�;
optBasisFn � getOpt�basisFn, opts, overlap�;
optShowProgress � getOpt�showProgress, opts, overlap�;
optConvertDomain� getOpt�convertDomain, opts, overlap�;
optFuncDomain� getOpt�funcDomain, opts, overlap�;
optBasisDomain� getOpt�basisDomain, opts, overlap�;
startX � optRange��1��;
endX � optRange��2��;
startY � optRange��3��;
endY � optRange��4��;
If �optShowProgress,Print�"Integrating �", n, ", ", m, "�"��;
�� Discrete function �� Use List integration ��
If �getOpt�discreteFunction, opts, overlap�,
�� First, compute a discrete representation of Subscript�e^�, n��x�Subscript�e^�, m��y�psi��x
are continuous, but the data function is already given as an array. ��
stepSizeX � �endX�startX��Dimensions�psi���1��;
stepSizeY � �endY�startY��Dimensions�psi���2��;
valueTable � Table�Conjugate�optBasisFn�m,convertDomain�optConvertDomain, startX�x�stepSizeX,
Conjugate�optBasisFn�n,convertDomain�optConvertDomain,startY � y�stepSizeY, optFuncDomain, optBasisDoma
�x, 1, Dimensions�psi���1��,1�, �y,1, Dimensions�psi���2��,1��;
�� Then perform the list integration. ��
Return�ListIntegrate2D�valueTable, �stepSizeX, stepSizeY���;
�;
�� Otherwise, proceed with regular numerical integration ��
Return�NIntegrate�Conjugate�optBasisFn�m,convertDomain�optConvertDomain, x, optFuncDomain, optBasisDoma
Conjugate�optBasisFn�n,convertDomain�optConvertDomain,y, optFuncDomain, optBasisDomain���� psi
�x, startX, endX�, �y,startY, endY�, AccuracyGoal��getOpt�accGoal, opts, overlap���;
�;
Options�overlap� � �basisFn��normH, showProgress��False, range����3,3,�3,3�, accGoal��4,
convertDomain��False, discreteFunction ��False,funcDomain����infinity,infinity�, basisDomain��

�� compute the Schmidt eigenmatrix �i.e., the matrix which is used to compute the eigenvalues
schmidtEM�F�, s�� :� Chop�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,s�, �j,1,s��
�� Compute the l’th �zero based� Schmidt function for overlap matrix F.
s denotes the maximal number of basis functions. ��
schmidtFnOne�F�, s�, l�, x�� :� schmidtFnOne�F, s, l, x� � Module��K,v,e�,
K � schmidtEM�F,s�;
v �Chop�Eigenvectors�K���l�1���;
Return�Sum�v��m���normH�m�1,x�, �m,1,s���;
�;
schmidtFnTwo�F�, s�, l�, x�� :� schmidtFnTwo�F, s, l, x� � Module��K,v,e�,
K �schmidtEM�F,s�;
v �Chop�Eigenvectors�K���l�1���;
e � Eigenvalues�K�;
Return�1�Sqrt�e��l�1����Sum�Conjugate�v��m����F��m,n���normH�n�1,x�, �m,1,s�, �n,1,s���;
�;
�� Here s denotes how many elements of the old basis are used to compute the new basis, i.e.
old basis functions a Schmidt basis function is composed of.
d tells how many of the new basis functions will be used, i.e., how many Schmidt terms are used
schmidtFn�F�, s�, d�, x�, y�� :� schmidtFn�F,s,d,x,y� � Module��K�,
Return�Simplify�Sum�Sqrt�schmidtEigenval�F,s���n�1����schmidtFnOne�F,s,n,x��schmidtFnTwo�F,s,
�;
�� Return the eigenvalues used to construct the Schmidt basis.
This measures how entangled the state is since the quantity is
invariant under basis transforms. Note that the number of basis elements
does not in the least imply anything about the basis dimension. ��
schmidtEigenval�F�, s�� :� Module��K�,
K � Chop�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,s�, �j,1,s���;
Return�Chop�Eigenvalues�K���;
�;
�� Distance measures between two functions f and g.
TODO: Adapt this to variable function domains, not only ��3,3�. ��
dist�f�, g�� :� NIntegrate�Abs�f�x,y� � g�x,y��^2, �x,�3,3�, �y,�3,3�,AccuracyGoal��4�;
distEV�f�,e�� :� 1��Sum�e��n��,�n,1,Length�e�����NIntegrate�Conjugate�f�x,y���f�x,y�, �x,�3,3
�� Some call it Shannon Entropy, others call it entropy of entanglement ��
S�l�� :� �Sum�l��n���Log�2,l��n���, �n,1,Length�l���;
Entanglement�F�, s�� :� S�Eigenvalues�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,
�� f is the function in the new polynomial basis ��not� the Schmidt basis��. T is the overlap
matrix, k denotes which order to use. ��
doSum�x�, y�,T�,k�� :� doSum�x,y,T,k�� Simplify�Sum�T��m,n���normH�m�1,x��normH�n�1,y�, �m,1,
f�x�, y�,T�,k�� :� doSum�x,y, T,k�;
�� Some sample functions. Gauss is, well, a Gau ian distribution, while shiftedGauss contains
gauss�x�, y�� :� gauss�x,y� � 1��2�Pi��Exp��0.5x^2�0.5y^2�;
gaussShifted�x�, y�� :� gaussShifted�x,y� � 1��2�Pi��Exp��0.5�x�1�^2�0.5�y�1�^2��1��2�Pi��Exp
�� TODO: Allow optional arguments be passed on to overlap ��
�� This computes a size x size overlap table for a given function ��
overlapTable�func�, size�� :� overlapTable�func, size� � Table�m,n,func, �m,0,size,1�, �n,0,size
EndPackage��
wmschmidt.m 3

A.2 id201 programming
The data acquisition of the id201 APDs has been fully automated, in addition to
the remote control of the external trigger and the APDs by the presented C++
program. It is based on previous work of Andreas Eckstein and has been modified
to communicate with the id201 APD and the Fluke trigger generator.
The program consists of four parts:
main.cpp specified the measurement and applied setups for id201 and Fluke.
fluke20.c.cpp communicated with the trigger generator and id201.cpp with the APD.
All data has been exchanged via RS232 and seriallink.cpp
100

������������������������������������������������������������������� �������������
�������������������
������������������
������������������
�������������������
�����������������
���������������������
������������������
��������������������
��������������������������
��������������������������������������
� ������������������������
�����������������������������������
����
���������������������������������������������
�������������������
������������������������������
����
������������������������������������
�����������
�����������������������������������
��
����������������������� ������������������������������
�������������������������� �������������������������
�������������������� ��������������������������������
������
������������������������
��������������������������
����������������������������������������
������������������������������
���� �������������������������������������
����
���� ����������������������������
���� �������������������������������������������
����
����
����
���� �
���� ������������������������������������������������������
����������������������������������
���� �
���� � �������������������������������
���� �
���������������
�����������������������������
���������������������������������������������
����������������������������������������������������������
����������������������������������������������������
����������������������������������������
������������������������������������������������������������
�����
���� ����������������������������������������
���� ������������������������
���� ���������

������������������������������������������������������������������� �������������
���� ������������������������
���� ���������
�������� ���������������������������������
�������� ��������������������������������
�����
�������� ���������� ������������������������������������������
���������������
��������
��������
���� ��
���� �����������������������������������
���� ��������������
�������
���� �
����
�����
������������������������������
�������������
�

(EXIM LSQI EGLVMWX 0EFSV MH MH û TVSKVEQQMIVYRK JPYOI \ GTT 7IMXI ZSR
MRGPYHI JPYOI \ L
MRGPYHI WWXVIEQ"
MRGPYHI YRMWXH L"
MRGPYHI WIVMEPPMRO L
MRGPYHI WXVMRK"
REQIWTEGI VIQSXIPEF
_
WXERHEVH GSRWXVYGXSV GVIEXIW JPYOI \ HIZMGI ZME
WIVMEP PMRO [MXL HIJEYPX GSRRIGXMSR TEVEQIXIVW
JPYOI \ JPYOI \ GSRWX WXVMRK HIZ VIQSXIHIZMGI RI[ WIVMEPPMRO HIZ @R@R _a
JPYOI \ JPYOI \ VIQSXIPMRO P VIQSXIHIZMGI P _a
JPYOI \ bJPYOI \ _a
FSSP JPYOI \ MRMX
_
FSSP SO
MJ PMRO MW3TIR
SO!PMRO STIR
a
MJ SO
VIXYVR JEPWI
VIXYVR XVYI
FSSP JPYOI \ WIX*VIUYIRG] PSRK J
_
SWXVMRKWXVIEQ SW
SW ;%:*6)5 J
GSYX WIRHMRK XS JPYOI SW WXV IRHP
PMRO WIRH'QH2S%RW[IV SW WXV
GSYX VIXYVR ZEPYI SYX IRHP
VIXYVR XVYI
a
a REQIWTEGI VIQSXIPEF

(EXIM LSQI EGLVMWX 0EFSV MH MH ûMH TVSKVEQQMIVYRK MH GTT 7IMXI ZSR
MRGPYHI MH L
MRGPYHI YRMWXH L"
MRGPYHI WWXVIEQ"
MRGPYHI WIVMEPPMRO L
MRGPYHI WXVMRK"
MRGPYHI EWWIVX L"
REQIWTEGI VIQSXIPEF
_
MH MH GSRWX WXVMRK HIZ VIQSXIHIZMGI RI[ WIVMEPPMRO HIZ @V _a
MH MH VIQSXIPMRO P VIQSXIHIZMGI P _a
MH bMH _a
FSSP MH MRMX
_
PEWX*VIUYIRG]!
a
FSSP SO
MJ PMRO MW3TIR
SO!PMRO STIR
;MV ZSR %4( RMGLX VMGLXMK ZIVEVFIMXIX /IMRI %LRYRK [EVYQ
GSYX 0MRO STIRIH 7IRHMRK XIWX GSQQERH XS ETH IRHP
WXVMRK SYX
PMRO WIRH'QH < SYX
GSYX XIWX GSQQERH VIWYPX SYX IRHP
MJ SO
VIXYVR JEPWI
VIXYVR XVYI
FSSP MH MW2SX6IJVIWLIH WXVMRK VIWYPX
_
MJ VIWYPX PIRKXL !!
VIXYVR JEPWI
a
MJ VIWYPX? A!!
VIXYVR XVYI
VIXYVR JEPWI
PSRK MH KIX*VIUYIRG]
_
XLMW JYRGXMSR QEOIW YWI SJ XLI PEWX*VIUYIRG] JMIPH XS WXSVI
XLI QIEWYVIH JVIUYIRG] ERH KMZI XLEX FEGO [LIR XLI JVIUYIRG] SJ
XLI %4( LEW RSX FIIR YTHEXIH ]IX
WXVMRK SYX
PMRO WIRH'QH ()8)'836 *6)59)2'=# SYX
MJ SYX PIRKXL !!
VIXYVR

(EXIM LSQI EGLVMWX 0EFSV MH MH ûMH TVSKVEQQMIVYRK MH GTT 7IMXI ZSR
a
MJ MW2SX6IJVIWLIH SYX
VIXYVR PEWX*VIUYIRG]
IPWI
_
MWXVMRKWXVIEQ MW SYX
MW "" PEWX*VIUYIRG]
VIXYVR PEWX*VIUYIRG]
a
FSSP MH WIX(IXIGXSV(IEHXMQI WXVMRK HIEHXMQI
_
WXVMRK SYX
MJ HIEHXMQI !! 232) `` HIEHXMQI !! `` HIEHXMQI !! `` HIEHXMQI !! ``
HIEHXMQI !!
_
WXVMRK MRTYX
MRTYX ! ()8)'836 (IEHXMQI HIEHXMQI
a
PMRO WIRH'QH MRTYX SYX
MJ SYX !! 3/
_
VIXYVR XVYI
a
IPWI VIXYVR JEPWI
a
IPWI VIXYVR JEPWI
FSSP MH WIX(IXIGXSV4VSFEFMPMX] WXVMRK TVSFEFMPMX]
_
WXVMRK SYX
MJ TVSFEFMPMX] !! `` TVSFEFMPMX] !! ``TVSFEFMPMX] !! ``TVSFEFMPMX]
!!
_
WXVMRK MRTYX
MRTYX ! ()8)'836 4VSFEFMPMX] TVSFEFMPMX]
a
PMRO WIRH'QH MRTYX SYX
MJ SYX !! 3/
_
VIXYVR XVYI
a
IPWI VIXYVR JEPWI
a
IPWI VIXYVR JEPWI
FSSP MH WIX(IXIGXSV;MHXL WXVMRK [MHXL
_
WXVMRK SYX
MJ [MHXL !! `` [MHXL !! ``[MHXL !! ``[MHXL !! `` [MHXL !!
_
WXVMRK MRTYX
MRTYX ! ()8)'836 ;MHXL [MHXL
PMRO WIRH'QH MRTYX SYX
MJ SYX !! 3/
_

(EXIM LSQI EGLVMWX 0EFSV MH MH ûMH TVSKVEQQMIVYRK MH GTT 7IMXI ZSR
a
VIXYVR XVYI
a
IPWI VIXYVR JEPWI
a
IPWI VIXYVR JEPWI
ZSMH MH WIX6IJVIWL6EXI WXVMRK VIJVIWL6EXI
_
WXVMRK SYX
MJ VIJVIWL6EXI !! ``VIJVIWL6EXI !! ``VIJVIWL6EXI !! ``VIJVIWL6EXI !!
``VIJVIWL6EXI !!
_
WXVMRK MRTYX
MRTYX ! (MWTPE] 6IJVIWL VIJVIWL6EXI
PMRO WIRH'QH MRTYX SYX
EWWIVX SYX !! 3/
a
a
a REQIWTEGI VIQSXIPEF

(EXIM LSQI EGLVMWX 0EFSV MH MH û TVSKVEQQMIVYRK WIVMEPPMRO GTT 7IMXI ZSR
MRGPYHI WIVMEPPMRO L
MRGPYHI YRMWXH L"
REQIWTEGI VIQSXIPEF
_
WIVMEPPMRO WIVMEPPMRO GSRWX WXVMRK HIZ GSRWX WXVMRK XIVQWXV TSVXCHIZMGI HIZ
XIVQ XIVQWXV
_
a
WIVMEPPMRO bWIVMEPPMRO
_
MJ 7IVMEP7XVIEQ -W3TIR
7IVMEP7XVIEQ 'PSWI
a
FSSP WIVMEPPMRO WIRH'QH GSRWX WXVMRK MR WXVMRK SYX
_
XLMW MR XIVQ
a
MJ XLMW
VIXYVR JEPWI
MRX FYJWM^I!
GLEV FYJ?FYJWM^IA
JSV MRX M! M FYJWM^I M
FYJ?MA!
WXVMRK XSOIR
GSYX KIXXMRK ERW[IV IRHP
KIXPMRI FYJ FYJWM^I @R
GSYX KSX ERW[IV IRHP
SYX ! WXVMRK FYJ
GSYX ERW[IV SYX IRHP
WXVMRK VIW
[LMPI XLMW
_
XLMW "" XSOIR
GSYX XSOIR XSOIR IRHP
VIW !XSOIR
a
VIXYVR XVYI
FSSP WIVMEPPMRO WIRH'QH2S%RW[IV GSRWX WXVMRK MR
_
XLMW MR XIVQ
a
VIXYVR XVYI
FSSP WIVMEPPMRO STIR
_
7IVMEP7XVIEQ 3TIR TSVXCHIZMGI
7IX&EYH6EXI 7IVMEP7XVIEQ&YJ &%9(C

(EXIM LSQI EGLVMWX 0EFSV MH MH û TVSKVEQQMIVYRK WIVMEPPMRO GTT 7IMXI ZSR
a
7IX'LEV7M^I 7IVMEP7XVIEQ&YJ ',%6C7->)C
7IX4EVMX] 7IVMEP7XVIEQ&YJ 4%6-8=C232)
7IX2YQ3J7XST&MXW
MJ KSSH
VIXYVR JEPWI
VIXYVR XVYI
FSSP WIVMEPPMRO MW3TIR
_
VIXYVR -W3TIR
a
ZSMH WIVMEPPMRO GPSWI
_
7IVMEP7XVIEQ 'PSWI
a
a

A.3 APD characterization
A.3.1 Dark counts
109

kcounts / second
kcounts / second
kcounts / second
kcounts / second
110
Gating width: 2.5 ns Detector efficiency 10%
0.025
0.02
0.015
0.01
0.005
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 20%
0.12
0.1
0.08
0.06
0.04
0.02
0.14
0.12
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.08
0.06
0.04
0.02
1.6
1.4
1.2
0.8
0.6
0.4
0.2
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 10%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 20%
1
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
kcounts / second
kcounts / second
kcounts / second
kcounts / second
Gating width: 2.5 ns Detector efficiency 15%
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0.14
0.12
0.1
0.08
0.06
0.04
0.02
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 25%
0.25
0.15
0.05
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.3
0.2
0.1
400
350
300
250
200
150
100
50
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 15%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]

A.3.2 1310 nm cw laser
The laser beam has been emitted from a LFO-14-ip laser diode. It is based on the
Mitsubishi 1310nm MQW InGaAs/InP Fabry-Perot laser diode and coupled to a
single mode fiber by Roitner GmbH.
kcounts / second
kcounts / second
kcounts / second
Gating width: 2.5 ns Detector efficiency 10%
3
2.5
2
1.5
1
0.5
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 20%
9
8
7
6
5
4
3
2
1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10
9
8
7
6
5
4
3
2
1
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 10%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
kcounts / second
kcounts / second
kcounts / second
Gating width: 2.5 ns Detector efficiency 15%
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
30
25
20
15
10
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
5
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
18
16
14
12
10
8
6
4
2
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 15%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
111

kcounts / second
80
70
60
50
40
30
20
10
Gating width: 5 ns Detector efficiency 20%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
A.3.3 1555 nm cw laser
kcounts / second
1800
1600
1400
1200
1000
800
600
400
200
Gating width: 5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
The laser beam has been emitted from a LFO-18-ip laser diode. It is based on the
Mitsubishi 1555nm MQW InGaAs/InP Fabry-Perot laser diode and coupled to a
single mode fiber by Roitner GmbH.
kcounts / second
kcounts / second
112
Gating width: 2.5 ns Detector efficiency 10%
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 20%
400
350
300
250
200
150
100
50
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
kcounts / second
kcounts / second
Gating width: 2.5 ns Detector efficiency 15%
140
120
100
80
60
40
20
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
800
700
600
500
400
300
200
100
trigger frequency [MHz]
Gating width: 2.5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]

kcounts / second
kcounts / second
350
300
250
200
150
100
50
Gating width: 5 ns Detector efficiency 10%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1600
1400
1200
1000
800
600
400
200
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 20%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
A.3.4 808 nm pulsed laser
kcounts / second
kcounts / second
800
700
600
500
400
300
200
100
Gating width: 5 ns Detector efficiency 15%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2000
1800
1600
1400
1200
1000
800
600
400
200
trigger frequency [MHz]
Gating width: 5 ns Detector efficiency 25%
Deadtime: 0 µs
Deadtime: 1 µs
Deadtime: 2 µs
Deadtime: 5 µs
Deadtime: 10 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
The laser beam has been emitted by the PicoQuant LDH-P-C-810 laser system.
kcounts per second
180
160
140
120
100
80
60
40
20
808nm pulsed laser 1nW, Width: 2.5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
kcounts per second
1800
1600
1400
1200
1000
800
600
400
200
808nm pulsed laser 1nW, Width: 5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
113

kcounts per second
kcounts per second
300
250
200
150
100
50
808nm pulsed laser 2nW, Width: 2.5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
500
450
400
350
300
250
200
150
100
50
114
808nm pulsed laser 4nW, Width: 2.5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
kcounts per second
kcounts per second
2000
1800
1600
1400
1200
1000
800
600
400
200
2000
1800
1600
1400
1200
1000
808nm pulsed laser 2nW, Width: 5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]
800
600
400
200
808nm pulsed laser 4nW, Width: 5 ns
Probability: 10%, Deadtime: 0 µs
Probability: 10% ,Deadtime: 1 µs
Probability: 10% ,Deadtime: 2 µs
Probability: 25%, Deadtime: 0 µs
Probability: 25%, Deadtime: 1 µs
Probability: 25%, Deadtime: 2 µs
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
trigger frequency [MHz]

A.4 SHG data
A.4.1 Different SHG processes in KTP
Our complete measurements to identify the different SHG processes, at different
wavelengths in KTP:
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
Pump central frequency: λ p = 1265 nm
SHG intensitiy [a.u.]
Pump central frequency: λ p = 1325 nm
SHG intensitiy [a.u.]
Pump central frequency: λ p = 1412 nm
SHG intensitiy [a.u.]
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
Pump central frequency: λ p = 1295 nm
SHG intensitiy [a.u.]
Pump central frequency: λ p = 1357 nm
SHG intensitiy [a.u.]
Pump central frequency: λ p = 1429 nm
SHG intensitiy [a.u.]
115

E y E y -> E y
E y E y -> E z
E y E z -> E y
E y E z -> E z
E z E z -> E y
E z E z -> E z
Pump central frequency: λ p = 1570 nm
SHG intensitiy [a.u.]
A.4.2 Chip BCT0703-B12
BCT0703-B12 first measurement
The complete data set of the first investigation of the BCT0703-B12 waveguide chip:
Shg power Pump power [a.u.]
116
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
2
1.5
1
0.5
Shg wavelength [nm] Pump wavelength [nm/2]
0
700 720 740 760 780 800 820 840
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
Shg wavelength [nm] Pump wavelength [nm/2]
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
2
1.5
1
0.5
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
2
1.5
1
0.5
Shg wavelength [nm] Pump wavelength [nm/2]
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
2
1.5
1
0.5
Shg wavelength [nm] Pump wavelength [nm/2]
Mapping pump wavelength to shg wavelength
Shg spectrum
Pump spectrum
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
117

BCT0703-B12 second measurement
The complete data set of the second investigation of the BCT0703-B12 waveguide
chip:
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
8
7
6
5
4
3
2
1
118
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
8
7
6
5
4
3
2
1
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]

Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
BCT0703-B12 third measurement
Shg power Pump power [a.u.]
8
7
6
5
4
3
2
1
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
The complete data set of the third investigation of the BCT0703-B12 waveguide
chip:
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
119

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
120
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]
121

Shg power Pump power [a.u.]
90
80
70
60
50
40
30
20
10
122
0
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
760 780 800 820 840
Shg wavelength [nm] Pump wavelength [nm/2]

A.4.3 Chip ITI0706-B12
ITI0706-B12 first measurement
The complete data set of the first investigation of ITI0706-B12 waveguide chip:
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
123

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
124
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
125

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
126
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]

Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
Shg power Pump power [a.u.]
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
30000
25000
20000
15000
10000
5000
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
Shg spectrum
Pump spectrum
Mapping pump wavelength to shg wavelength
0
700 720 740 760 780 800 820 840
shg wavelength [nm] / pump wavelength [nm/2]
127

ITI0706-B124 power measurement
The missing data of the second investigation of ITI0706-B12 waveguide chip:
137

SHG Efficiency (a. u.)
SHG Efficiency (a. u.)
138
SHG in the KTP waveguide Λ = 104.17 mum Pump Power = 5 µW
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450
λp [nm]
SHG in the KTP waveguide Λ = 104.17 mum Pump Power = 40 µW
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450
λp [nm]
SHG Efficiency (a. u.)
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
SHG in the KTP Λ = 104.17 mum Pump Power = 10 µW
0
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450
λp [nm]

Bibliography
[1] H. J. Kimble, M. Dagenais, and L. Mandel. Photon antibunching in resonance
fluorescence. Phys. Rev. Lett., 39(11):691–695, Sep 1977.
[2] J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning. Demonstration
of an all-optical quantum controlled-not gate. Nature, 426:264, 2003.
[3] Chao-Yang Lu, Xiao-Qi Zhou, Otfried Guhne, Wei bo Gao, Jin Zhang, Zhen
sheng Yuan, Alexander Goebel, Tao Yang, and Jian-Wei Pan. Experimental
entanglement of six photons in graph states. Nature Physics, 3:91, 2007.
[4] Charles Santori, David Fattal, Jelena Vuckovic, Glenn S Solomon, and Yoshihisa
Yamamoto. Single-photon generation with inas quantum dots. New Journal of
Physics, 6:89, 2004.
[5] M Hennrich, T Legero, A Kuhn, and G Rempe. Photon statistics of a nonstationary
periodically driven single-photon source. New Journal of Physics,
6:86, 2004.
[6] M Keller, B Lange, K Hayasaka, W Lange, and H Walther. A calcium ion in a
cavity as a controlled single-photon source. New Journal of Physics, 6:95, 2004.
[7] T Gaebel, I Popa, A Gruber, M Domhan, F Jelezko, and J Wrachtrup. Stable
single-photon source in the near infrared. New Journal of Physics, 6:98, 2004.
[8] W E Moerner. Single-photon sources based on single molecules in solids. New
Journal of Physics, 6:88, 2004.
[9] Jeremie Fulconis, Olivier Alibart, Jeremy L. O’Brien, William J. Wadsworth,
and John G. Rarity. Nonclassical interference and entanglement generation
using a photonic crystal fiber pair photon source. Physical Review Letters,
99(12):120501, 2007.
[10] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn,
and I. A. Walmsley. Heralded Generation of Ultrafast Single Photons in
Pure Quantum States. Physical Review Letters, 100(13):133601–+, April 2008.
139

[11] P. J. Mosley, J. S. Lundeen, and B. J. Smith. Conditional preparation
of single photons using parametric downconversion: A recipe for purity.
arXiv:0807.1409v1, 2008.
[12] Philippe Grangier, Barry Sanders, and Jelena Vuckovic. Focus on single photons
on demand. New Journal of Physics, 6, 2004.
[13] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich. Generation of optical
harmonics. Phys. Rev. Lett., 7(4):118–119, Aug 1961.
[14] Robert W. Boyd. Nonlinear Optics Second Edition. Academic Press, 2003.
[15] David C. Burnham and Donald L. Weinberg. Observation of simultaneity in
parametric production of optical photon pairs. Phys. Rev. Lett., 25(2):84–87,
Jul 1970.
[16] Rodney Loudon. The Quantum Theory of Light. Oxford University Press, 3
edition, 2000.
[17] Alfred B. U’Ren. Multi-photon State Engineering for Quantum Information
Processing Applications. PhD thesis, Universitiy of Rochester, 2004.
[18] Leonard Mandel Emil Wolf. Optical coherence and quantum optics. Cambridge
University Press, 1995.
[19] D. Marcuse. Theory of dielectric optical waveguides. New York, Academic
Press, Inc., 1974. 267 p., 1974.
[20] Thomas Lauckner. Modale effekte in wellenleitern und zwei-photonenfluoreszenz.
Master’s thesis, Universität Erlangen-Nürnberg, 2008.
[21] Alfred B. U’Ren, Christine Silberhorn, Reinhard Erdmann, Konrad Banaszek,
Warren P. Grice, Ian A. Walmsley, and Michael G. Raymer. Generation of
pure-state single-photon wavepackets by conditional preparation based on spontaneous
parametric downconversion. Laser Physics Letters, 15:146, 2005.
[22] Toshiaki Suhara and Masatoshi Fujimura. Waveguide Nonlinear-Optic Devices
(Springer Series in Photonics, V. 11). SpringerVerlag, 2003.
[23] Mark C. Booth, Mete Atature, Giovanni Di Giuseppe, Alexander V. Sergienko,
Bahaa E. A. Saleh, and Malvin C. Teich. Counter-propagating entangled photons
from a waveguide with periodic nonlinearity. Physical Review A, 66:023815,
2002.
[24] Jun Chen, Kim Fook Lee, and Prem Kumar. Deterministic quantum splitter
based on time-reversed hong-ou-mandel interference. Physical Review A
(Atomic, Molecular, and Optical Physics), 76(3):031804, 2007.
140

[25] Carlot Canalias and Valdas Pasiskevicius. Mirrorless optical parametric oscillatro.
Nature Photonics, 1:4, 2007.
[26] Anthony N. Vamivakas, Bahaa E. A. Saleh, Alexander V. Sergienko, and
Malvin C. Teich. Theory of spontaneous parametric down-conversion from
photonic crystals. Phys. Rev. A, 70(4):043810, Oct 2004.
[27] Malte Avenhaus. Photonenstatistik von parametrischer fluoreszenz. Master’s
thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg Max-Planck-
Nachwuchsgruppe für Integrierte Quantenoptik Max-Planck-Forschungsgruppe
am Institut für Optik, Information und Photonik, 2007.
[28] H B Coldenstrodt-Ronge and C Silberhorn. Avalanche photo-detection for high
data rate applications. Journal of Physics B: Atomic, Molecular and Optical
Physics, 40(19):3909–3921, 2007.
[29] Jr. Jan Peˇrina. Quantum properties of counterpropagating two-photon states
generated in a planar waveguide. Physical Review A (Atomic, Molecular, and
Optical Physics), 77(1):013803, 2008.
141

142

Thanks
❏ To Dr. Christine Silberhorn, who gave me the opportunity to work in the
IQO-group, her constant support and guidance.
❏ To Andreas Eckstein, my mentor, for hours of time and his enduring patience.
❏ To Thomas Lauckner whose calculations crucially supported this thesis.
❏ To Malte Avenhaus who helped me to test and build the Mathematica packages.
❏ To Wolgang Maurer for deep insights into Quantum Optics, Mathematica and
LaTeX
❏ To Kaisa Laiho who supplied me with experimental data.
❏ To Christoph Söller who kept the lasers smoothly running.
❏ To Wolfram Helwig for his help with the Schmidt decompositions.
❏ To Felix Just, who helped me with the SHG experiments.
❏ To the whole IQO-group, Christine Silberhorn, Andreas Eckstein, Kaisa Laiho,
Malte Avenhaus, Christoph Söller, Wolfgang Maurer, Thomas Lauckner, Wolfram
Helwig, Felix Just, Benjamin Brecht and Andreas Schreiber, for a superb
working climate and all the fun in the past year.
❏ Division 3 and the NONA group for lending us lab equipment.
❏ The workshop team Bernhard Thomann and Robert Gall, they always supplied
me with the necessary tools and parts to perform our experiments.
143

144

Erklärung
Gemäß §31(2) der Diplomprüfungsordnung für Studenten der Physik an der Friedrich-Alexander-Universität
Erlangen-Nürnberg vom 20.08.2004 versichere ich, dass
ich die Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen
und Hilfsmittel benutzt habe.
Erlangen, 1.August 2008
Andreas Christ
145